AN ABSTRACT OF THE THESIS OF
Heeyong Park for the degree of Doctor of Philosophy in
Mechanical Engineering presented on December 12. 1991.
Title: A Study of Laser Generated Rayleigh and Lamb Waves
in Graphite/Epoxy Composites.
Abstract Approved :
Redacted for Privacy
Clarence A. Calder
The application of laser generated ultrasonics was first
demonstrated in the mid-seventies and has shown good potential
when applied to isotropic materials.
However, its use with
composite materials is still in the early stages of development.
This study explores the potential for application of laser
generated Rayleigh and Lamb waves in graphite/epoxy composites.
Numerical results are obtained by the solution of the wave equations
using assumed solutions, and enforcing the boundary conditions.
Experimentally, Rayleigh and Lamb waves were generated by a Q-
switched ruby laser in the ablation regime and detected by piezo-
electric pinducers which permitted accurate phase velocity
measurements.
The Rayleigh wave velocity was measured at various directions
relative to the fiber direction and results were found to agree closely
with numerical predictions.
The increase of surface wave velocity
using thin plates could be useful for the application of delamination
detection in thick composites and an increase of Rayleigh wave
attenuation could indicate damages caused by impact.
waves can reflect from small surface cracks.
Also, surface
Therefore, laser
generated surface waves, particularly along the fiber direction, have
high potential for application in non-destructive testing.
Lamb wave experiments were conducted in aluminum plates
and gave distinctive signals, but there were some difficulties in
detecting the precise arrival of each Lamb wave mode for the
graphite/epoxy composite plates.
A Study of Laser Generated Rayleigh and Lamb Waves
in Graphite/Epoxy Composites.
by
Heeyong Park
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirement for the
degree of
Doctor of Philosophy
Completed December 12, 1991
Commencement June 1992
APPROVED:
Redacted for Privacy
Associate Professor of Mechanical Engineering in charge of major
Redacted for Privacy
Head of Department of Mechanical Engineering
Redacted for Privacy
c
Dean of Graduate
6
hool
q
Date thesis is presented
December 12, 1991
Typed by researcher for
Heeyong Park
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to Professor C. A.
Calder, for his academic guidance and financial support as my advisor
and his encouragement like my father in Korea.
I appreciate my
thesis committee members, T. C. Kennedy, M. N. L. Narasimhan, E.
Wolff, and Graduate Committe Representative, R. H. Cuenca, for their
support and Wyle Labs in Edwards Air Force Base for graphite/epoxy
specimens.
Patience, support, and love of my wife ( Young-mee Park ) and
my father ( Hae-chul Park ) are beyond description.
like to share my Ph.D with my mother in heaven.
Finally, I would
TABLE OF CONTENTS
1. INTRODUCTION
1.1. General
1.2. Literature Review
1.2.1. Rayleigh Wave
1.2.2. Lamb Wave
1.3. Purpose of Study
2. BACKGROUND
2.1. The Potential of Laser-generated Ultrasound
2.2. Generation of Ultrasonic Waves by Laser Deposition
2.2.1. Absorption of Laser Energy
2.2.2. Thermoelastic Effects of the Surface
2.2.3. Ablation of the Surface
2.3. Michelson Interferometer
3. NUMERICAL SOLUTIONS FOR RAYLEIGH AND LAMB
WAVES
3.1. Formulation of Equations for Rayleigh Waves
3.1.1. Wave Equation
3.1.2. Assumed Solutions for Rayleigh Waves
3.1.3. Secular Equation for Rayleigh Waves
3.1.4. Boundary-condition Determinant
3.2. Numerical Solutions of Rayleigh Waves
3.2.1. Numerical Procedure
3.2.2. Material Constants
3.2.3. Numerical Results
3.3. Formulation of Equations for Lamb Waves
3.3.1. Assumed Solutions for Lamb Waves
3.3.2. Secular Equation of Lamb Waves
3.3.3. Boundary-condition Determinant
3.4. Numerical Solutions of Lamb Waves
3.4.1. Numerical Procedure
3.4.2. Numerical Results
1
1
3
3
6
10
11
11
13
13
15
17
18
23
23
23
24
26
27
29
29
32
32
43
43
44
45
48
48
50
4. EXPERIMENTS
4.1. Experimental Arrangement
4.2. Preparation of Specimen
4.3. Experimental Procedure
4.4. Experimental Results and Discussion
4.4.1. Rayleigh Waves
4.4.2. Lamb Waves
5. CONCLUSIONS AND RECOMMENDATIONS
55
55
61
61
64
64
75
82
BIBLIOGRAPHY
84
APPENDIX A : PROGRAM FOR RAYLEIGH WAVES
93
APPENDIX B : PROGRAM FOR LAMB WAVES
101
111
APPENDIX C : COMPONENTS OF [Aii]
APPENDIX D : EXPERIMENTAL WAVEFORMS AND
FFT ANALYSIS OF RAYLEIGH WAVES
112
APPENDIX E : EXPERIMENTAL WAVEFORMS AND
FFT ANALYSIS OF LAMB WAVES
117
LIST OF FIGURES
Figure
Page
Thermoelastic effects of the surface
at low laser power density.
17
2.2.2.
Ablation of the surface at high laser power density.
18
2.3.1.
Basic Michelson interferometer.
19
2.3.2.
Sensitivity comparison of Michelson interferometer.
22
3.1.1.
Coordinate system for the Rayleigh wave propagation.
25
3.2.1.
Finding three roots of L3 from an eigen value plot.
30
3.2.2.
Flow chart for numerical search procedure of
2.2.1.
3.2.3.A
3.2.3.B
3.2.4.A
3.2.4.B
3.2.4.0
3.2.4.D
Rayleigh wave velocity.
31
Polar plot of Rayleigh and body wave phase velocities
in the unidirectional graphite/epoxy composite.
34
Rayleigh and body wave phase velocities in the
unidirectional graphite/epoxy composite as a function
of the wave propagation direction.
34
Normalized X,Y, and Z displacements vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite (0 = 0° ).
36
Normalized X,Y, and Z displacements vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite ( 0 = 10° ).
36
Normalized X,Y, and Z displacements vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite (0 = 30° ).
37
Normalized X,Y, and Z displacements vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite (0 = 90° ).
37
3.2.5.
3.2.6.
3.2.7.A
3.2.7.B
3.2.7.0
3.2.7.D
3.3.1.
3.4.1.
Angular deviation at the surface between Rayleigh
wave propagation directions and horizontal
displacement vectors given every 10°.
39
Change of horizontal displacement vector at every
0.02 wave length depth relative to the Rayleigh wave
propagation direction.
40
Normalized X,Y, and Z stresses vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite ( 0 = 0° ).
41
Normalized X,Y, and Z stresses vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite ( 0 = 30° ).
41
Normalized X,Y, and Z stresses vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite ( 0 = 60° ).
42
Normalized X,Y, and Z stresses vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite ( 0 = 90° ).
42
Coordinate system for the Lamb wave propagation
43
Combined eigen value plot of [Aid] according to the
L3 value.
49
Lamb waves propagation in the unidirectional
graphite/epoxy composite plates ( 0 = 0° ).
51
Lamb waves propagation in the unidirectional
graphite/epoxy composite plates ( 0 = 45° ).
51
Lamb waves propagation in the unidirectional
graphite/epoxy composite plates ( 0 = 90° ).
53
3.4.3.
Lamb waves propagation in the aluminum plate.
53
3.4.4.
The Ao mode velocities by the change of wave
propagation directions in the unidirectional
graphite/epoxy composite plates.
54
3.4.2.A
3.4.2.B
3.4.2.0
3.4.5.
4.1.1.
4.1.2.
4.1.3.
4.3.1.
4.4.1.
The So mode velocities by the change of wave
propagation directions in the unidirectional
graphite/epoxy composite plates.
54
Schematic diagram of experimental setup
for the single pinducer technique.
56
Schematic diagram of experimental setup
for dual pinducer technique.
57
Measurements of Rayleigh wave propagation time
by the single and the dual pinducer techniques.
60
Layout of laser deposition tests for Rayleigh and
Lamb waves generation.
62
Rayleigh waves in the unidirectional graphite/
epoxy composite as a pinducer moves from
a line source (0 = 0° ).
4.4.2.
4.4.3.
4.4.4.
66
Rayleigh waves in the unidirectional graphite/
epoxy composite as a pinducer moves from
a line source ( 0 = 90° ).
66
Relation between the laser energy and
the pinducer output.
67
Attenuation of Rayleigh wave amplitudes
in the graphite/epoxy composite
4.4.5.
4.4.6.
4.4.7.A
when 0 = 0° & 90° ( laser energy = 205 mJ ).
70
Calculations and experiments of Rayleigh wave
phase velocities in the unidirectional graphite/
epoxy composite and comparison with Rose's
results [see reference 57].
72
Comparison between the single and the dual
pinducer techniques in Rayleigh wave velocity
measurements.
74
Laser generated ultrasonic waves in an aluminum
plate of 0.8 mm thickness.
76
4.4.7.B
Frequency analysis of Fig. 4.4.7.A.
76
4.4.8.A
Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
perpendicular to the fiber direction.
78
4.4.8.B
Frequency analysis of Fig. 4.4.8.A.
4.4.9.
Digital filtering analysis for Lamb wave signal of
78
Fig. 4.4.8.A
i) Low pass filtering with a cut-off frequency
of 0.7 MHz,
ii) Band pass filtering between 0.7 and 1.7 MHz,
iii) High pass filtering with a cut-off frequency
of 1.7 MHz.
4.4.10.
79
Polar plot of surface wave phase velocities
on the graphite/epoxy composite plates of
thickness 0.87 & 16.25 mm.
80
LIST OF TABLES
Table
3.2.1.
Page
Material constants of graphite/epoxy specimen
with 60 % fiber volume fraction ( T300/5208 ).
32
4.2.1.
Dimensions of graphite/epoxy plates ( T300/5208 ).
61
4.4.1.
Attenuation calculation of a graphite/epoxy composite.
69
LIST OF APPENDIX FIGURES
Figure
D.1.A
Page
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 16.25 mm thickness ( 0 = 0° ).
112
D.1.B
Frequency analysis of Fig. D.1.A.
112
D.2.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 16.25 mm thickness ( 0 = 20° ).
113
D.2.B
Frequency analysis of Fig. D.2.A.
113
D.3.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 16.25 mm thickness ( 0 = 50° ).
114
D.3.B
Frequency analysis of Fig. D.3.A.
114
D.4.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 16.25 mm thickness ( 0 = 70° ).
115
D.4.B
Frequency analysis of Fig. D.4.A.
115
D.5.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 16.25 mm thickness ( 0 = 90° ).
116
D.5.B
Frequency analysis of Fig. D.S.A.
116
E.1.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 0.87 mm thickness ( 0 = 0° ).
117
E.1.13
Frequency analysis of Fig. E.1.A.
117
E.2.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 0.87 mm thickness ( 0 = 20° ).
118
E.2.B
Frequency analysis of Fig. E.2.A.
E.3.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
118
plate of 0.87 mm thickness ( 0 = 40° ).
119
E.3.B
Frequency analysis of Fig. E.3.A.
119
E.4.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 0.87 mm thickness ( 0 = 50° ).
120
E.4.B
Frequency analysis of Fig. E.4.A.
120
E.5.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 0.87 mm thickness ( 0 = 60° ).
121
E.5.B
Frequency analysis of Fig. E.S.A.
121
E.6.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
plate of 0.87 mm thickness ( 0 = 80° ).
122
E.6.B
Frequency analysis of Fig. E.6.A.
122
E.7.A
Laser generated ultrasonic waves
in the unidirectional graphite/epoxy composite
E.7.B
plate of 0.87 mm thickness ( 0 = 90° ).
123
Frequency analysis of Fig. E.7.A.
123
A STUDY OF LASER GENERATED RAYLEIGH AND LAMB WAVES
IN GRAPHITE/EPDXY COMPOSITES
CHAPTER 1
INTRODUCTION
1.1.
General
When the equations of motion for an infinite isotropic solid
are solved, it is found that there are two types of elastic body waves
that can propagate independently: 1) the longitudinal wave in which
the particle motion is parallel to the direction of propagation and
2) the transverse wave in which the motion is perpendicular to the
direction of propagation.
If the medium occupies a half-space and
has one free surface, then the solutions must satisfy the boundary
condition that the surface stresses are zero.
This condition can be
used to find the reflection coefficients for body waves, but a solution
exists that describes a wave confined to the near-surface. Waves of
this type are referred to as Rayleigh waves.
They propagate along
the free surface and decay exponentially in the direction normal to
the surface.
In a plate, there are two surfaces at which traction-free
boundary conditions must be satisfied.
When the boundary
conditions are imposed, the Rayleigh-Lamb equations relating the
wave number, K ( = 27c/X, ), and the angular frequency co are obtained.
2
It is found that, for a given value of K, there is an infinite number of
frequencies which satisfy the Rayleigh-Lamb equations.
These
correspond to different modes of Lamb waves.
Ultrasonic waves including Rayleigh and Lamb waves have been
one of the most powerful tools of NDE ( Non-Destructive Evaluation )
for isotropic materials.
However, the attention of the NDE
community has shifted toward composites because their very large
strength-to-weight and stiffness-to-weight ratios are attractive for a
wide range of applications, especially in the aerospace industry.
Many of the NDE tools available for testing of isotropic materials have
been applied to composites.
Quite naturally, ultrasonic testing
has been used for the NDE of composites with varying degrees of
success.
The major difficulty in case of composites arises from the
fact that the theoretical analysis of wave propagation is considerably
more difficult.
For example, in an isotropic material, the wave
propagation and energy propagation directions are the same: In an
anisotropic material, however, they are quite different in general.
The most popular tool of ultrasonic NDE has been the
conventional piezoelectric transducer and the various, associated
techniques have been developed.
There are, however, limitations
of piezoelectric transducers such as required physical contact,
temperature limits, flat surface for coupling, etc.
Therefore,
NDE of composites by laser generated stress waves with detection
using various interferometry techniques along with fiber optics is
currently under active development.
3
1.2.
Literature Review
1.2.1. Rayleigh Wave
Lord Rayleigh established the basic properties of acoustic
surface wave propagation along the surface of an elastic isotropic
solid.
He showed that the wave is non-dispersive, propagates at a
velocity slightly less than the shear wave velocity, and that most of
its energy is contained within a wave length of the surface [54].
It was realized during the 1950's that Rayleigh waves could be useful
for nondestructive testing.
Much of the early work in this field
was carried out in Russia and Germany.
Major contributions on
the application of Rayleigh waves to NDT ( non-destructive testing
were made by Victorov [73].
)
He pointed out that Rayleigh waves
in the ultrasonic range could be used to detect the presence of flaws
such as cracks and holes, near the surface of a sample.
Also, he
pointed out that the change in attenuation and velocity of Rayleigh
waves could be used as an indication of the material state near the
surface of a solid.
An important problem of great interest in structural mechanics
Surface cracks weaken
is the determination of the size of a crack.
a material and lead to its eventual fracture.
If it is assumed that
the Rayleigh wavelength is large compared to the size of the crack,
then it is possible to develop quasi-static theories for Rayleigh wave
scattering from a crack based on fracture mechanics theories.
The basic
Typically, surface cracks have a half-penny shape.
quantitative theory to determine these effects for Rayleigh waves was
4
developed by Kino [40].
An NDT technique for larger cracks, which
has been highly successful, was carried out by Silk [63].
He made
use of the fact that a surface wave incident on a crack propagates
along the surface of substrate, then along the surface of the crack,
and radiates as a body wave from the tip of the crack.
Detection of
this body wave can be obtained on the top or bottom surfaces of the
substrate, away from the crack.
In either case, a good estimation
of the crack depth can be made by measuring the extra time delays
due to propagation of the wave along the length of the crack.
In 1963 White demonstrated laser generation of acoustic waves
in a solid [75].
The first models describing the phenomenon were
one-dimensional [76,55,12,74], and assumed that the laser impact
occupied the entire surface of the material.
Since the laser
generated ultrasound was received by means of narrow passband
piezoelectric transducers, harmonic directivity patterns were
initially used for describing the ultrasonic waves generated by
a point laser impact [35,36].
Subsequently the use of broadband
receivers such as capacitive sensors [21,62], thick piezoelectric
discs [22], or laser interferometers [69,38,8] have made wideband
modelling of laser generated acoustic displacements necessary.
Laser generated ultrasound is now widely accepted as a versatile
NDT technique in the laboratory, and is starting to find industrial
applications [61,56,6,39,59,60,2].
Cooper, Dewhurst, and Palmer
studied interaction of laser generated Rayleigh pulses with surface
breaking slots in metal using a capacitance probe [20].
Surface
5
echoes reflected from the defect had two major components.
The
first arose from direct reflection of a Rayleigh pulse from the top of
the defect, whereas the second arose from a shear pulse originating
at the bottom of the defect which mode-converted to a Rayleigh
pulse on reaching the metal's surface at the critical angle [23].
On the other hand, fiber optic sensor systems for ultrasonic
NDT have been developed by many authors [24,25,9,17].
Fiber
optic sensors have some advantages over conventional transducers
and laser interferometers.
For example, a flexible fiber can
reach rather inaccessible surfaces and can be readily incorporated
into a scanning system.
Duffer, Burger, and Piper machined a
slot on a steel specimen and used a dual channel fiber tip interfero-
meter to detect the Rayleigh wave as it passed underneath each
sensor [26].
Comparison of the frequency spectrum of the
incident Rayleigh wave with that of the transmitted Rayleigh waves
contained information about the depth of surface cracks.
Recently, Huang and Achenbach employed a dual-probe
interferometer to obtain accurate measurements of the surface wave
forms on an aluminum plate with increasing degrees of surface
roughness [33].
McKie et al designed and developed a dual-beam
interferometer for the accurate measurement of surface wave
velocity on an aluminum block [49] .
A system of polarizing optical
components was used in order to efficiently derive two beams from
a single input laser beam.
By accurate Rayleigh wave measurement,
the degree of anisotropy of the test specimen could be obtained.
6
Development of laser-based ultrasonics for composites is now
in the early stages.
Addison, Jr., Ryden, and McKie made
measurements of the angular diffraction pattern for longitudinal
elastic waves from a laser generated thermoelastic source in both
aluminum and a graphite/epoxy composite [1].
Tittman, Linebarger,
and Addison, Jr. demonstrated a laser based transmission C-scan of
a simulated ( 10 mm by 10 mm ) delamination in a graphite/epoxy
composite [70].
1.2.2. Lamb Wave
The governing equations for Lamb waves were first derived
by Professor Horace Lamb in 1917 in his famous work [44].
He
formulated the problem using potentials and arrived at the wellknown Rayleigh-Lamb equation for wave propagation in isotropic
plates.
These equations were quite complicated and a solution
could be obtained only in the short and long wavelength limits.
The first comprehensive solution of Lamb waves was obtained by
Mindlin [50] in 1950.
Later, Viktorov dealt with the solution of
Lamb waves in great detail [73].
He provided the dispersion
curves for aluminum with a Poisson's ratio of 0.34.
Solie and Auld
investigated waves in anisotropic plates theoretically and compared
the results to the uncoupled shear vertical and longitudinal modes
[64].
Habeger, Mann, and Baum conducted a theoretical and
experimental study of ultrasonic Lamb waves in machine-made
7
papers [30].
The papers were modeled as homogeneous
orthotropic plates.
The dispersion equation for Lamb wave
propagation in the principal directions was developed analytically
and verified experimentally.
Moon used a variational method
to obtain an approximate solution for Lamb wave propagation in
laminated composite plates [51].
Sun and Tan derived an
approximate solution based on Mind lin's plate theory [68].
Stiffler and Henneke obtained a low-frequency Lamb wave solution
using elementary plate theory [67].
Mal developed a general
theory, based on a matrix formulation, to solve the wave propagation
problem in multi-layered composite laminates [46], and the results
have been corroborated with experimental data by Mal and BarCohen [47,5].
There are many methods to generate and detect Lamb waves in
composite materials.
wave method.
The most popular method is the leaky Lamb
In this particular method, two transducers and a
composite plate are immersed in a water tank.
By varying the
sending and receiving angles of the transducers, relative maxima
corresponding to Lamb modes are detected when the Lamb wave
energy leaks from the surfaces of the plate.
By knowing the angles
and the excitation frequencies, dispersion curves can be obtained
experimentally.
Worlton was the first one to confirm the theory of
Lamb waves and determined the dispersion curves for aluminum and
zirconium experimentally using this method [77].
Chimenti and
Nayfeh verified their approximate solution for unidirectional
8
composite plates in the fiber direction by the detection of null zones
that indicated the presence of leaky Lamb waves [19].
Bar-Cohen
and Chimenti studied the application of leaky Lamb waves for NDT of
composite laminates [4].
They observed a correlation between
the character of the excited Lamb wave modes and the presence of
certain defects such as porosity and delaminations.
Martin and
Chimenti refined this method by signal processing [48].
The results
of the leaky Lamb wave method look promising, but the technique
may not be particularly well suited for field inspection of composite
laminates because the method requires the plates to be immersed in
water.
A simple method similar to the acousto-ultrasonic technique
[72,32] has been studied by Stiff ler and Henneke to generate and
detect Lamb waves in composite laminates [67,27,66].
The experi-
mental data obtained by this simple method generally belong to the
lowest symmetric and anti-symmetric Lamb modes.
In this method,
two piezoelectric transducers acoustically coupled to the plate by
water-soluble couplant are directly in contact with the composite
plate to create and receive Lamb waves.
The Lamb wave speed can
be determined by measuring the change in arrival time of a phase
point on the receiving signal while moving the receiver a certain
distance.
Chapman used the same idea in the detection of
debonding in fiberglass-reinforced plastic lap joints [18].
In a
similar approach, Rose, Rokhlin, and Adler measured Lamb wave
speeds with energy flux deviation in composite laminates [58].
9
Liu generated Lamb modes and SH modes in a unidirectional
graphite/aluminum composite plate by a shear wave transducer and
used an electro-magnetic acoustic transducer ( EMAT ), which was
not sensitive enough to detect the signals [45].
Wormley and
Thompson, who used two EMATs as receivers, determined the wave
speed by cross-correlation between the two receiving signals and
thus evaluated the texture of rolled-metal plates [78].
Recently, laser-generated Lamb waves were studied by
Hutchins and Lundgren [37].
The Lamb wave was propagated
within thin materials ( aluminum and metallic glass samples ) and
detected by Michelson interferometer.
The thickness of the
samples and their elastic constants were estimated with good
accuracy.
Burger et al generated Lamb waves in a thin aluminum
plate by Nd:YAG laser and detected signals by a fiber optic system
[10].
They demonstrated good agreement between experiment
and finite element models.
Aussel and Monchalin measured the dispersion of Rayleigh
waves on thick and thin metallic substrates using laser-ultrasonics.
In order to enhance the precision of the measurements, the signalto-noise ratio was increased by focusing the Rayleigh wave with an
annular generating source [3].
Hutchins et al presented the
ultrasonic inspection of composite panels, fabricated by attaching a
thin layer of aluminum to a rigid foam substrate using an epoxy
resin, and of aluminum composites consisting of aluminum sheets
bonded together with an epoxy adhesive layer [34].
A ruby laser
10
beam was focused to a line source and the receiver was an EMAT for
the detection of vertical displacements.
They demonstrated that
the presence of high frequencies serves as a test for disbond.
1.3. Purpose of Study
To date, laser ultrasonics have been developed mainly for
isotropic materials as given in the literature review.
Recently,
it has been used for applications in composite materials, which are
highly anisotropic.
The characteristics of laser generated stress
waves including Rayleigh waves, Lamb waves, and body waves need to
be fully explored and understood so that NDE of composite materials
by laser systems will soon be possible.
In this study the characteristics of laser generated Rayleigh
waves and Lamb waves was studied for unidirectional graphite/epoxy
composite plates.
Graphite fibers with an epoxy matrix
constitute one of the most popular composites.
The velocity
changes of Rayleigh and Lamb waves according to wave propagation
direction was studied numerically and verified experimentally as far
as possible.
Also, the attenuation of Rayleigh waves and their
dependence on wave propagation direction and on the distance
between laser source and detection point was studied for future
NDE applications using graphite/epoxy and similar composites.
11
CHAPTER 2
BACKGROUND
2.1. The Potential of Laser-generated Ultrasound
By far the most commonly used method of generating and
detecting ultrasound has been by piezoelectric transducer.
Historically, piezoelectric crystals such as quartz were
predominantly used as transducer materials.
Many of the problems with piezoelectric generation and
reception lie not in the transducer, but in the coupling which is
necessary between transducer and specimen.
The choice of
couplant ( often an oil or grease ) and its method of application often
tend to be an art rather than a science.
Variability of couplant
thickness and partial transmission and partial reflection of the
ultrasonic energy in the couplant layer result in loss of sensitivity,
change of waveform, unwanted resonances and time-of-flight
measurement errors.
The entire test structures are often
immersed in a tank of water or coupling fluid.
Although the
couplant allows acoustical energy to propagate into the test material,
it causes several problems in addition to potential harm to the
material, particularly porous ceramics or polymer-based materials.
There are additional problems and limitations of using piezoelectric
transducers [60,61].
The laser generation of ultrasound can
eliminate the need for any coupling between source and sample and
overcomes all of these problems.
12
Calder and Wilcox showed the potential of laser-generated
ultrasound through the following developments:
1) demonstration
of the detection of an artificial flaw, a 1.5 mm diameter hole located
midway across a 25 mm aluminum plate [13];
2) the measurement of
dilatational and mode-converted pulse arrivals in rods along with
calculation of the elastic constants of many isotropic materials [14,15];
3) the use of laser energy deposition and wave detection by Michelson
interferometer to provide a unique method for the measurement of
acoustic velocity in liquid metals [16].
They pointed out some distinct advantages of laser ultrasonics
over the conventional transducers:
1) The laser loading produces a large stress pulse amplitude of
short duration so that tests of highly attenuating or very thin
materials are possible.
2) The short measurement time of a few microseconds or less
can be made at a precise instant in time.
3) The noncontact feature is especially useful for testing in severe
environments and with toxic materials.
4) There are few restrictions on specimen size and configuration.
( Laser beams can be focused to a point or a line by optical
methods. )
5) The specimen can be far removed from the instrumentation
hardware.
13
The following sections will give the background of how
ultrasonic waves are generated by laser deposition and detected by
Michelson interferometer.
2.2. Generation of Ultrasonic Waves by Laser Deposition
2.2.1. Absorption of Laser Energy
A pulsed laser emits bursts of coherent electro-magnetic
radiation. When low intensity radiation ( 106 W/cm2) is incident on
a metal surface, the combination of electric and magnetic fields
generates currents in the conduction band near the surface of the
material.
Some of the incident energy is absorbed by resistive
losses and converted into heat, while the remainder is re-radiated
as a reflected pulse. Most of the absorption and reflection takes
place very close to the surface, within what is called "skin depth" 8.
At longer wavelengths in the infrared, the following classical
expression for 5 can be used [7]
S2
1
Eafligo
(2.2.1)
where a = conductivity of material
g = relative permeability of material
f = frequency of the incident radiation
P.O = 4n x 10-7 H/m = permeability of free space.
14
For a Nd:YAG laser of wavelength 1.0611m in the near infrared, the
skin depth is - 5 nm in aluminium.
Again,using classical electromagnetic theory [7) the reflectivity
of the material R can be calculated as the ratio of the reflected
energy Er to the incident energy Ei i.e.,
R=
Er
2-2t+t2
=
(2.2.2)
2+2t+t2
where t = 1.10005
c = velocity of light.
For a metal, t »1 for all frequencies up to visible light, so that
reflectivity can be approximated as
4
R=1-T
(2.2.3)
.
Therefore, the absorbed energy Ea is given by
Ea =
(1 R)
=
(2.2.4)
For example, theoretical absorption ratio is 6% for aluminium.
Thus in a low power regime where other effects can be
neglected, the pulsed laser acts as a transient heat energy source at
the surface.
If the absorption of energy is assumed to occur so
rapidly that thermal conductivity into the bulk of the material can be
neglected, the deposition of energy Ea in a surface layer of area A
and depth 8 causes a temperature rise sr given by
AT =
Ea
spAS
(2.2.5)
15
where s = specific thermal capacity of material
P = density of material.
Substituting Ea from eq (2.2.4) gives
4
AT =
ilocpase
(AJ
(2.2.6)
Therefore the temperature rise is proportional to the mean incident
energy density.
Over the timescales of typical laser pulses ( 10 ns to 100 ns ),
there is, in fact, appreciable thermal conductivity into the bulk of
most materials [55,76].
The main effects as regards the generation
of ultrasound are that the volume of heated material increases as a
function of time.
2.2.2. Thermoelastic Effects of the Surface
The sudden rise in temperature of the surface layer by the
absorption of thermal energy should be accompanied by thermal
expansion.
If the volume expands from v.M to V + AV, then
AV = 3aVAT
(2.2.7)
where a = coefficient of linear expansion.
Substituting for AT from eq (2.2.5) and for Ea from eq (2.2.4) gives
AV =
3a
sp
E =
a
3a
sp
(1-R) Ei
(2.2.8)
16
Therefore, the absorption of laser energy causes thermoelastic
strains equivalent to the sudden insertion of a volume AV of material
immediately below the surface that is proportional to the incident
energy Ei.
Note that the magnitude of the increased volume is
independent of both the beam area A and the thickness 8 at
constant reflectivity R.
If the source is situated deep in the
material, it will appear to be a pure expansion and will generate only
compressional waves in all direction. However, the close proximity
of the actual source to the surface introduces some conversion of
the compressional waves to shear waves [11].
As shown in Fig.
2.2.1.A, compressive dipolar stresses parallel to the surface are
unchanged, and the boundary conditions require zero net stresses
normal to the surface.
However, since the heating occurs to the
skin depth, the source is actually just below the surface so that a
small normal stress can be produced.
This thermoelastic source
of ultrasound is different from a piezo-electric compressional
transducer which principally generates a stress normal to the
surface.
For many applications it is adequate to treat the thermo-
elastic source of ultrasound as a point or line source, provided
sufficient beam focusing is employed.
When the surface is
constrained by oil layer, glass, plastic cover, etc. as shown in Fig.
2.2.1.B, the stress normal to the surface becomes dominant and
is the cause of dramatic increase in the generation efficiency of
longitudinal waves.
17
Incident
Incident
Laser Pulse
Laser Pulse
Skin
Depth
Constraining
5
- - el.
Layer
40- --
-P.
ik.
Dipolar Surface Stress
(A) Free Surface
Normal stress due to
Thermal Expansion
(B) Constrained Surface
Fig. 2.2.1 Thermoelastic effects of the surface
at low laser power density.
2.2.3. Ablation of the Surface
From eq (2.2.6.), the surface temperature increases linearly
with incident energy or power density.
As the optical power
density is increased, whether by increasing the energy per pulse
or by focusing a constant energy onto a small surface area using a
converging lens, the temperature must rise until the melting point of
the surface material is reached and ablation takes place from the
surface.
Once ablation occurs, some of the incident energy is
dissipated as kinetic energy of the material vapor.
During transient
heat pulses of short duration, thermal equilibrium is not reached, so
that the surface can become superheated for a short time above the
18
boiling point of the material.
The detailed interactions among the
laser pulse, plasma, and surface are beyond the scope of the present
discussion [431.
When the laser intensity is sufficient to cause ablation, the
impulse force given to the surface by the vaporization of coating
material or ablation of the surface itself produces a strong impulsive
recoil force, by the transfer of momentum, as with a constrained
surface in the thermoelastic region ( refer to Fig. 2.2.2 ).
M
Laser Pulse
Momentum due to
Surface Vaporization
iii
-> Dipolar Surface Stress
due to Thermal Expansion
Normal Stress due to
Momentum Transfer
Fig. 2.2.2 Ablation of the surface at high laser power density.
2.3. Michelson Interferometer
As described in sec. 2.1, laser-generated ultrasound has great
potential in NDE.
Many types of interferometric techniques [28,651
19
can be useful for noncontact, remote sensing, optical detection.
The Michelson interferometer shown in Fig. 2.3.1 is the basic one
which has been widely used.
The basic principle of fiber optic
detection is also the same as that of the Michelson interferometer.
Therefore, the following discussion on the principles of Michelson
interferometry will also be useful for understanding other laser
based interferometric methods.
Fig. 2.3.1 shows a schematic for this interferometer.
from a laser is divided by a beam splitter into two parts.
Light
One part
goes through the beam splitter and strikes a movable reflecting
surface at a distance Xi from the beam splitter.
The other part of
beam is reflected by the beam splitter to a fixed reference mirror
located at a distance X2 from the beam splitter and is reflected back.
Reference Mirror
Specimen
Beam
Splitter
Out of plane motion
Detector
(Photomultiplier Tube)
Fig. 2.3.1 Basic Michelson interferometer.
20
The two beams are recombined at the beam splitter and travel
Because these beams originate from the
together to the detector.
same coherent source, sharp interference fringes are produced at
the detector which represent a measure of the difference in the
Note that the optical path
optical paths of the two beams [531.
length changes twice as much as the out of plane motion of the
For large displacements, it is necessary to count
fringes since each fringe indicates an optical path difference of only
specimen.
X/2.
For changes less than X/2 , a measure of the variation in the
fringe intensity gives the change in position, since the fringe
intensity varies sinusoidally as a function of optical path difference.
For very small displacements, the interferometer can be stabilized
on the most sensitive position of the sine wave resulting in an
output that is nearly linear with displacement.
The light intensity Lt measured at the detector of an
interferometer is
Lt = [ IC, Re exp { i (KX coot + 2KX1(t ) + 01(t ))}
± 4172 Re exp { i (KX
coot + 2KX2(t ) + 02(t )) }]2 (2.3.1)
where co. = 27cf. (fo = optical frquency )
= intensity of the object beam (i=1) or reference beam (i=2)
K = 2rc / X = wave number
cOi(t ) = noise fluctuations in object beam (i=1) or ref. beam (i=2).
Since all the contributions to the signal from extraneous thermal or
mechanical fluctuations are contained in 41(t) and (02(t ), X2 can be
21
considered constant and the variation of X1(t) from its equilibrium
value ( X1(0)) is the variable of interest, e.g. AX, (t ) = X1(t) X1(0).
Then, eq (2.3.1) can be written
2 1.
1 = L1 +L2 F O r T 2 cos ( 2KAX,(t ) + A4)(t ) +43 )
where octo = K [ Xi (0)
(2.3.2)
X2(0) l,
04) .-. 4)1(t ) _ 4)2(t )
= noise displacement.
Obviously, 4) is a system parameter and can be adjusted to any
convenient value, i.e. it / 2.
Therefore, after removing electronically
the constant term from eq (2.3.2), the signal is given by
Ls =
II7I72 sin ( 2KAXs(t )+ A4)(t ))
(2.3.3)
Fig. 2.3.2 shows the sensitivity change of interferometer.
The sine
function is most sensitive to changes in As when the argument is
small ( or - nn ,where n = integer ).
is directly proportional to AXs.
Then, the measured intensity
However, if 2ICAX2(t )+ A4)(t ) - it / 2
( or (n + 1) 7E / 2 ), the interferometer is operating at its least sensitive
position and relatively large changes in AXs cause little change in
the observed intensity.
Because, typically, 2KAXs(t ) « 1 and
dA4)(t ) / dt « dAXs(t)/dt, the noise A4)(t) causes the signal to drift in
and out of the most sensitive region.
This noise results in low
frequency shift of the output signal, which changes signal sensitivity
and makes the system unstable.
It is important to compensate for
this noise to maintain the interferometer at its most sensitive
position [6].
22
Most Sensitive
Case
Least Sensitive
Case
2 kAXs
2 IcAXs
Optical Phase
Fig. 2.3.2. Sensitivity comparison of Michelson interferometer.
23
CHAPTER 3
NUMERICAL SOLUTIONS FOR RAYLEIGH AND LAMB WAVES
3.1. Formulation of Equations for Rayleigh Waves
3.1.1. Wave Equation
In a perfectly elastic, homogeneous, anisotropic medium without
body forces and piezoelectric effects, the wave equation can be written
by using the usual summation convention,
a2 Ui
P Tat
a2 Uk
corm
axiax,
(3.1.1)
where Ui = displacement component along the Cartesian axes Xi
to which the stiffness tensor Ciikm is referred
P = density of the medium
j, k, m = 1, 2, 3.
Here, the epoxy matrix of the graphite/epoxy composite is assumed
to be elastically isotropic and the fibers are assumed to be transversely isotropic.
If fibers are distributed randomly in the matrix,
then the graphite/epoxy can have hexagonal symmetry.
Of course,
this is not homogeneous in a microscopic point of view.
However,
if the wave length of the Rayleigh wave is large with respect to the
diameter of fibers ( the diameter of the graphite fibers is about 5 lim),
then the scattering by the individual fibers can be ignored and the
graphite/epoxy composite can be treated as homogeneous and
transversely isotropic.
24
If the medium is infinite in all directions, the simplest
solutions of eq (3.1.1) are given by the real part of
U = A exp[iK(LiXi
with A = ii (xi
vt)]
(3.1.2)
= 1, 2, 3), where ii is a unit vector along the Xi axis and K
is a wave number. The phase velocity v of the wave is measured along
the propagation vector K, whose direction cosines are given by the Li
(K=
ii).
These homogeneous plane waves are called bulk waves.
In general, there will be three distinct velocities of body wave
propagation [421.
The values of these velocities will depend on the 21
elastic constants of the material, and on the direction of propagation.
3.1.2. Assumed Solutions for Rayleigh Waves
The coordinate system for the Rayleigh wave problem will use
X3 as the outward normal to the traction-free surface of the medium
as shown in Fig. 3.1.1.
The axis X1 is chosen in the convenient
direction, e.g. along the fiber direction.
Solutions for Rayleigh waves are assumed to decay with depth
below the surface and to be homogeneous plane waves whose
displacements are independent of the distance perpendicular to the
wave propagation direction. Therefore, the solutions are assumed to
be linear combinations of terms in the form
Ui = ai exp[iK(LiXi - vt )]
= ai exp[iKL3X3] exp[iK(Li XI + L2X2
(3.1.3)
vt )]
25
which satisfy the wave equation (3.1.1) and the traction-free surface
condition.
Conceptually, the term oci exp[iICL3X3] is regarded as the
"amplitude" of the solution which varies in the X3 direction over the
planes of constant phase.
The planes of constant phase are taken to
be perpendicular to the surface and to the propagation vector.
The
next term is taken as the wavelike properties. Thus, the propagation
vector is always assumed to be parallel to the surface so that L1 and L2
are assumed to be real and can be initially defined.
(outward normal
to the surface)
X3 (Z)
Surface
of material
Fiber direction
A
..........
L1= COS°
L2 = sine
Plane of Constant Phase
K Propagation Ve ctor
Fig. 3.1.1. Coordinate system for the Rayleigh wave propagation.
26
The quantity 1,3 should be such that the amplitude of all the
displacement components vanish as X3 ---> -00.
In other words,
L3 with a negative imaginary part can only satisfy the boundary
condition of zero displacement at infinite depth.
3.1.3. Secular Equation for Rayleigh Waves
The assumed solutions of the Rayleigh waves should satisfy the
wave equation (3.1.1).
Substituting eq (3.1.3) into eq (3.1.1), the
following homogeneous set of equations are obtained:
Ajk
8jk PV2
aj = 0
where Ajk = LiLinCijkm (i, j, k, m = 1, 2, 3).
(3.1.4)
Details are given in
appendix C.
In order to have nontrivial solutions, it is necessary that the
determinant of the coefficients be zero,
Ajk
8jk PV2
=0.
(3.1.5)
This secular equation can be regarded as a cubic equation in v2
with L3 as a parameter and the phase velocity v can be calculated from
the eigenvalue of tensor A.
Or the equation can be considered as a
sextic equation in L3 with v as a parameter.
For any specified value
of v, any root L3 of eq (3.1.5) gives a solution of eq (3.1.3).
In a
general anisotropic material, there are three pairs of complex
27
conjugate roots in this equation for each value of v. If the wave is
propagating on the plane of material symmetry, eq (3.1.5) becomes
bicubic due to material symmetry. However, the roots L3 with
negative imaginary part are taken to satisfy the boundary condition as
discussed in the previous section 3.1.2.
In case of isotropic
materials, all roots lie on the imaginary axis of the complex plane.
Therefore, three roots of
I.z3
should be combined together for
the assumed solution to eq (3.1.3) because they satisfy eq (3.1.5) and
each term has the same phase velocity v.
The assumed solution is
3
= ± Wn ap) exp[ iK(Li XI + L2X2 +L(3n)X3
vt)]
(3.1.6)
n=1
a.(n)= thee components of the eigenvector of eq (3.1.4)
according to the root Ij3n)
Wn = weighting factor.
These three weighting factors Wn should be determined by the
satisfaction of traction-free surface boundary conditions.
3.1.4. Boundary-condition Determinant
The boundary conditions at the surface are given as follows.
T3j = C3jkm
auk
axm =
at X3 = 0
(j=1,2,3)
(3.1.7)
28
Substituting eq (3.1.6) into the boundary conditions eq (3.1.7) and
setting X = 0 and t = 0 in order to omit the constant term,
3
T3i
C3 jkm Wn a(kn) i K 1(n11)
(3.1.8)
n =1
3
Ir
,c3Jkm 4,1) Om° Wni K =0
n=1
where L(in) = L1 and L2) = L2 for notation convenience.
Therefore, three linear homogeneous equations from the surface
traction-free boundary condition are given as follows.
B. W. = 0
1.1
(i, j = 1, 2, 3)
(3.1.9)
where Bij = C3ikm ak
In order to get a nontrivial solution, the determinant of the
coefficient should be zero.
IBij
I =0
(3.1.10)
It should be noted that the eigenvectors a(k) can be complex here.
The latter determinant " Boundary-condition Determinant " is
complex in general [291.
29
3.2. Numerical Solutions of Rayleigh Waves
3.2.1. Numerical Procedure
As discussed in section 3.1.3., there are three pairs of complexconjugate roots to eq (3.1.5) for each assumed value of phase velocity
in a general anisotropic material.
These roots L3 should satisfy eq
(3.1.4).
In other words, one of the eigenvalues of [
,
v
(i, j = 1, 2, 3)
should be equal to pv2 for any assumed value of v and propagation
direction. But the correct eigenvalue can be found only when wave
propagation direction is parallel or perpendicular to the fiber
direction.
Otherwise, the coupling of equations leads to calculation
errors during L3 root finding resulting in incorrect eigenvalues.
In order to get the correct eigenvalues and eigenvectors, a root
search method other than direct root finding will be used.
In
Fig.3.2.1, only half of eigenvalue plot is given and [ Ai) has three
eigenvalues for each negative imaginary value of L3.
In the case of
an isotropic material, two of three eigenvalues are equal, i.e.,
degenerated.
Here, "limit eigenvalue" EL can be defined as the
smallest eigenvalue at L3 = 0 and "limit velocity" VL can be given by
EL.
This means that the Rayleigh wave velocity VR is limited by
0 < V R < VL
VL,
(3.2.1)
where VL = 4/TTP .
Also, the limit velocity of an isotropic material is the shear velocity.
For an assumed phase velocity v, there are three intersections
between three eigenvalue plots and the "assumed eigenvalue" pv2
30
which is less than EL . Therefore, the root L3 can be found by a
suitable search algorithm for each eigenvalue.
Root Root
Root
Fig. 3.2.1. Finding three roots of L3 from an eigenvalue plot.
The Rayleigh wave velocity can be found when the boundary
condition determinant (BCD) is equal to zero as follows ( refer to Fig.
3.2.2.):
1) The search interval of the phase velocity should be carefully
selected from a rough plot of BCD.
Wide interval selection can
result in an wrong value of phase velocity because of the abrupt
change in the BCD value.
2) The interval is reduced until the specified tolerance is met.
The tolerance should be small enough to make both the real
and imaginary part value of the BCD close enough to zero.
If
the tolerance is not met, return to step 1 and select the other
interval.
31
start
)
.LIM111
Plot rough BCD value
Reduce search interval
by root finding algorithm
BCD value
is close enough to
zero ?
Yes
Plot displacements
Decay of
displacements
with depth?
Yes
Try
Another?
End
Fig. 3.2.2. Flow chart for numerical search procedure
of Rayleigh wave velocity.
32
3) After the velocity calculation, the displacement plot should be
checked to make sure that displacements decay with depth.
Otherwise, an incorrect velocity can be obtained.
3.2.2. Material constants
Typical material constants of graphite/epoxy are given in Table
3.2.1 [52,41].
Only 5 material constants are needed for a transversely
isotropic material and Tsai's index convention is used [71].
Table 3.2.1. Material constants for graphite/epoxy specimen with
60 % fiber volume fraction ( T300/5208 ).
Density
p
1.52 g / cm3
Poisson's ratio
v 21
0.31
v23
0.54
E1
138.5 GPa
E2
11.0 GPa
G12
6.3
Young's modulus
Shear modulus
GPa
3.2.3. Numerical results
For given material constants, the phase velocity of the
Rayleigh wave can be calculated by the numerical procedure given in
33
section 3.2.1.
Also, there are three body waves in anisotropic
materials as discussed in section 3.1.1: one quasi-longitudinal wave
( QL ) and two quasi-transverse waves ( QT ), which can be calculated
from eq (3.1.4).
Note that these body waves generally do not have
particle displacements which are purely parallel or perpendicular to
the wave propagation direction.
Therefore, the body waves of
anisotropic materials are generally neither longitudinal nor
transverse as those of isotropic materials [31].
If three velocities
are different, eq (3.1.4) implies that the vibration directions
corresponding to the three velocities are mutually perpendicular.
In Fig. 3.2.3.A and B, the QL wave always shows the highest wave
velocity.
The QL wave is about four times faster than the QT wave
when wave propagation is parallel to the fiber direction.
This
results from the high Young's modulus along the fiber direction.
For wave propagation perpendicular to the fiber, the QL wave
velocity approaches the value of the two QT wave velocities.
As expected, Rayleigh wave velocity is below the limit velocity
corresponding to the slowest body wave.
Rayleigh wave velocity
has its maximum 2.02 ( km/sec ) at 0 = 0°, and is continuously
decreased to its minimum 1.44 ( km/sec ) at 0
= 90°.
Displacements are given by the real part of the complex
solution of eq (3.1.6).
Note that the X axis is parallel to the fiber
direction and the Z axis is outward normal to the surface in Fig. 3.1.1.
The X and Y displacement solutions have only real parts and the Z
displacement solution has only an imaginary part.
This means that
34
10
Body wave QL, QT
Rayleigh wave
VELOCITY (KM/SEC), PARALLEL TO FIBER
Fig. 3.2.3.A Polar plot of Rayleigh and body wave phase velocities
in the unidirectional graphite/epoxy composite.
10
__
Body wave
Rayleigh wave
------- ------------------------------------- _
.............r-a-na,-..,
----------------------------------------------
10
20
30
40
50
60
70
80
90
ANGLE RELATIVE TO THE FIBER DIRCTION (DEGREE)
Fig. 3.2.3.B Rayleigh and body wave phase velocities in the
unidirectional graphite/epoxy composite as a function
of the wave propagation direction.
35
the Z displacement is ic/2 phase shifted relative to the X and
displacements.
Y
Therefore, for comparison with the X and Y
displacements, the Z displacement is plotted with an imaginary part.
From Fig. 3.2.4.A to 3.2.4.D, the right end of abscissa is the surface of
material and the left end is the inside of the material.
Note that the
absolute value of the calculation has no importance, and X, Y, and Z
displacements are normalized relative to the surface Z displacement.
There is no Y displacement when 0 = 0° and no X displacement when
0 = 90°, because of material property symmetry.
High stiffness of
the fiber makes X displacement relatively small and almost zero at a
depth lower than 0.1 wave length.
The Y displacement direction
changes between the surface and a depth of about 0.2 wave length.
The Z displacement shows a maximum value essentially at the surface
and disappears at a depth of about 5 wave lengths.
From 0 = 30° to
overall displacement patterns are similar and show only minor
changes of amplitude. When 0 = 90°, the displacement pattern of a
90°
,
unidirectional graphite/epoxy is exactly the same as that of isotropic
materials because the plane of wave propagation is transversely
isotropic.
Isotropic materials have no horizontal transverse displacement
component of the Rayleigh wave.
However, the unidirectional
graphite/epoxy has a horizontal transverse displacement component
and there is an angular deviation between the wave propagation
direction and the horizontal displacement vector.
When 0 = 30°,
angular deviation at the surface is about 25.7° as shown in Fig. 3.2.5.
36
1.2
X disp.
00000
Y disp.
Z disp.
0.8
0.6
0.4
0.2
0.2
5
4.5
4
3
3.5
2.5
2
1.5
1
0.5
0
DEPTH (Z) / WAVE LENGTH
A) 0 = 0°
1.2
X disp.
00000
1
Y disp.
Z disp.
0.8
0.6
0.4
0.2
0.2
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
DEPTH (Z) / WAVE LENGTH
B ) 0 = 10°
Fig. 3.2.4. Normalized X,Y, and Z displacements vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite
37
1.2
X disp.
00000
1
Y disp.
Z disp.
0.8
0.8
0.4
0.2
0
0.2
5
4.5
4
3.5
3
2.5
1.5
2
1
0.5
0
DEPTH (Z) / WAVE LENGTH
C ) 0 = 30°
1.2
1
0.8
0.6
0.4
0.2
0
0.2
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
DEPTH (Z) / WAVE LENGTH
D)
= 90°
Fig. 3.2.4. Normalized X,Y, and Z displacements vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite
38
At increasing value of depth Z, this deviation becomes larger toward
the Y axis and the X displacement approaches zero as shown in Fig.
The high Young's modulus along the X axis minimizes the X
3.2.6.
displacement inside the material resulting in the X displacement
rapidly reducing to near zero at about 0.1 wave length depth.
Stresses can be easily calculated using generalized Hooke's law
given as
T1 = Cijk,
aDUk
(i, j = 1, 2, 3)
(3.2.2)
The three normal stresses disappear at a depth of about three wave
lengths.
T11 ( ax ) is dominant when 0 = 0°, but T22 (6y ) is dominant
when 0 = 90°.
The stress plots shown in Fig. 3.2.7.A to 3.2.7.D were
normalized relative to the surface T22 for comparison and T11 can be
regarded as the average value assuming homogeneity of the graphite/
epoxy composite because the axial normal stress T11 of the fiber is
much higher than that of the matrix at the same elongation.
39
2
Wave Propagation direction
Displacement Unit Vector
1.8
1.6
E-
1.4
1.2
1
w
a.
0.8
4.1
0.6
0.4
rn
0.2
0
0
0.5
1
1.5
2
0 DEGREE (FIBER DIRECTION)
Fig. 3.2.5. Angular deviation at the surface between Rayleigh
wave propagation directions and horizontal
displacement vectors given every 10°.
40
L6
c:4
1.4
Displacement Unit Vector
Wave Length
1.2
E1
0.8
a.,
1:1-
0.6
44
0.4
A
rn
0.2
0
0
0.5
1
15
0 DEGREE (FIBER DIRECTION)
Fig. 3.2.6. Change of horizontal displacement vector at every
0.02 wave length depth relative to the Rayleigh wave
propagation direction.
41
60
50
0 0 0 0
40
6Z
30
20
10
.;
0
-10
-5
-4.5
-4
-3.5
-3
31,,
-2
-2.5
-1.5
711,111
-1
-0.5
0
DEPTH (Z) / WAVE LENGTH
A ) 0 = 0°
16
14
12
0000
10
8
6
4
2
0
2
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-0.5
0
DEPTH (Z) / WAVE LENGTH
B ) 0 = 30°
Fig. 3.2.7. Normalized X,Y, and Z stresses vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite
42
ax
3.5
a
0 0 0 0
aZ
2.5
1.5
0.5
----------
-0.5
-5
-4
-4.5
-3.5
-3
-2
-2.5
-1.5
-1
-0.5
DEPTH (Z) / WAVE LENGTH
C) 0 = 60°
1
ax
0.8
0 0 0 0
a
aZ
0.6
0.4
0.2
0
-0.2
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
DEPTH (Z) / WAVE LENGTH
D) 0 = 90°
Fig. 3.2.7.
Normalized X,Y, and Z stresses vs. depth plot
for Rayleigh wave propagation in the unidirectional
graphite/epoxy composite
43
3.3. Formulation of Equations for Lamb Waves
3.3.1. Assumed Solutions for Lamb Waves
Lamb waves occur in relatively thin plates.
The coordinate
system for the Lamb wave problem here will be similar to that of the
Rayleigh wave problem with X3 (Z) as the outward normal to the
traction-free surface of the medium as shown in Fig. 3.3.1.
X3 (Z)
(outward normal
to the surface)
Fiber direction
Ll = COS°
L2 = sine
N----11°'
Plane of Constant Phase
K Propagation Vector
X
Fig. 3.3.1 Coordinate system for the Lamb wave propagation.
44
The axis X1 (X) is chosen along the fiber direction and
X2 (Y)
perpendicular to the fiber direction.
As before, the propagation
vector H lies at 0 degrees from the X1 (X) axis.
As with the Rayleigh wave solution, the solution for the Lamb
wave is assumed to be linear combinations of the terms of the form
U = A exp[iK(LiXi
vt )]
(3.3.1)
which satisfy the wave equation (3.1.1) and the traction-free surface
boundary conditions.
The propagation vector is always assumed to be parallel to the
surface ( L1 and L2 are assumed to be real and can be initially
defined ).
Displacements do not necessarily vanish at the bottom
surface similar to Rayleigh waves in the semi-infinite domain and
can have any amplitude as long as eq (3.1.1), eq (3.3.1), and the
traction-free surface boundary conditions are satisfied.
Therefore,
the quantity L3 can have either a positive or negative imaginary part.
3.3.2. Secular Equation for Lamb Waves
The assumed solutions of the Lamb wave should satisfy the
wave equation (3.1.1).
Substituting eq (3.3.1) into eq (3.1.1), the
following equation is obtained as found previously ( eq.3.1.4 ),
( Ajk
8jk pv2 ) aj = 0
(3.3.2)
45
where Ajk =LiLmCiikm (i, j, k, m = 1, 2, 3).
In order to have a nontrivial
solution, it is necessary that the determinant of the coefficients be
zero,
I Aik
8jk pv2 I = 0
(3.3.3)
.
This secular equation is also the same as that of the Rayleigh wave.
Therefore, six roots of L3 should be combined together for the
assumed solution eq (3.3.1) because they satisfy eq (3.3.3) and each
term has the same phase velocity v. The assumed solution is
I
6
tii =
wn 41) exp[ iK(Li Xi + 11X2 +II3n)X3
vt)]
(3.3.4)
n =1
where a.( `)= the components of the eigen vector of eq (3.3.2)
according to the root L(3n)
Wi, = weighting factor.
These three weighting factors Wn should be determined by the
satisfaction of traction-free surface boundary conditions.
3.3.3. Boundary-condition Determinant
With the enforcement of boundary conditions, the solution
procedure is similar to that for the Rayleigh wave.
The boundary
conditions at the top and bottom surfaces are given as follows.
46
auk
°
(j = 1, 2, 3 )
(3.3.5)
T3i = C3jkm axm = °
(j= 1,2,3)
(3.3.6)
T3j = C3jkm axm
at X3 =0
auk
at X3 = -h
Substituting eq (3.3.4) into the boundary conditions eq (3.3.5) and
setting X = 0 and t = 0 in order to omit the constant term,
6
E C3ikmwvvn ak
T3i
1K
(n)
(3.3.7)
n=1
6
E [C3jkm
=
Lm61) Wni K = 0
at X=0 &t=0
n =1
where Li = L1 and L2(n) = L2 for notation convenience
.
Again, substituting eq (3.3.4) into the boundary conditions eq (3.3.6)
and setting X1 = 0 = X2 & t = 0 in order to omit the constant term,
6
E
T3i =
(n)
(n)
C3jkm %lc ak
i KLm
exp(-iKL3() h)
(3.3.8)
n =1
6
E [C3jkm okn)
=
exp(-iKLV h)] Wni K = 0
n=1
at X = (0,0,-h) & t=
Therefore, six linear homogeneous equations from eq (3.3.7) and eq
(3.3.8) result from the surface traction-free boundary conditions and
are given as
47
Dij Wi = 0
(
j = 1, ..., 6)
where Dm" = C3mki C1") 11")
(3.3.9)
for m = 1, 2, 3
Dnin = C3 6,1_3 )ki a(11,1) L(in) exp(-iK041) h)
for m = 4, 5, 6
k, j = 1, 2, 3.
In order to get a nontrivial solution, the determinant of the
coefficients should be zero.
I
Dmn
l
=0
(3.3.10)
The latter determinant is the "Boundary-condition Determinant" for
Lamb wave propagation.
48
3.4.
Numerical Solutions of Lamb Waves
3.4.1. Numerical procedure
The search method for Lamb wave solutions is similar to that
of the Rayleigh wave except for minor differences.
Rayleigh wave
velocity has an upper limit, i.e., limit velocity VL, and L3 can have
negative imaginary values only.
The Lamb wave velocity is not
restricted by VI, and L3 can have a real or pure imaginary value
depending on the assumed velocity.
Therefore, a modified search
method is used for evaluation of Lamb wave velocities.
L3 is real
for solid lines and is pure imaginary for dotted lines in Fig. 3.4.1.
There are three intersections E1, E2, E3 of solid and dotted lines at
L3 = 0.
Four different search regions of L3 are divided as follows.
Region I
: 0 < pv2 < Ei
Region II
: El < pv2 < E2
,
4 imaginary roots & 2 real roots
Region III
: E2 < pv2 < E3
,
2 imaginary roots & 4 real roots
Region IV
: E3 < pv2
6 imaginary roots
6 real roots
The numerical search procedure given in Fig. 3.2.2 can also be used
for the Lamb wave velocity calculation, but the boundary value
determinant of the Lamb wave is different from that of the Rayleigh
wave and a modified search method of L3 should be used. Instead
of checking decay of displacement with depth, the displacement
pattern should be a symmetric or anti-symmetric mode as given in
the next section.
49
X10-5
E3
E2
El
Eigenvalues with imaginary L3
Eigenvalues with real L3
-1.5
1
0.5
0
0.5
1
15
L3 or L3/i
Fig. 3.4.1. Combined eigen value plot of [Aid] according to the
L3 value.
50
3.4.2. Numerical Results
The Lamb wave has a different phase velocity depending on
the propagation direction as with the Rayleigh wave but is dispersive
unlike the Rayleigh wave. The phase velocity v of the Lamb wave is
related to wave number K, wave length X, period T, and frequency co
as follows:
v=
co
K
= 27r
- X- =
X,
T 27c
(3.4.1)
T
For a dispersive wave, the relationship between phase velocity and
wave length X is nonlinear. There are an infinite number of
solutions satisfying eq
and eq (3.3.10) for the given values of
plate thickness h and wave length X. They can be divided into two
(3.3.3)
groups as for Lamb waves in isotropic materials.
When the vertical
Z displacement is symmetric about the central plane of the plate, it
is called the " symmetric mode".
If the Z displacement is antisymmetrical about the central plane of the plate, it is called the
"anti-symmetric mode" corresponding to flexural vibrations of the
plate.
First two symmetric modes So, S1 and anti-symmetric modes
Ao, Al are given for the cases of 0
= 0°, 45°,
90°.
When 0 = 0°, the Ao
mode velocity is continuously increasing and approaches the
Rayleigh wave velocity as ha increases as shown in Fig. 3.4.2.A.
The So mode velocity approaches the QL wave velocity when the
plate is very thin (11/A 0) and approaches the Rayleigh wave velocity
as the plate becomes thicker.
When 0 = 45° in Fig.
3.4.2.B,
the So
51
10
-
9
SYMMETRIC MODE
ANTISYM. MODE
8
7
6
2
5
4
O
3
2
1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.8
2
PLATE THICKNESS ( h ) / WAVE LENGTH
A) 0 = 0°
7
SYMMETRIC MODE
6
ANTISYM. MODE
5
4
3
2
1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
PLATE THICKNESS ( h ) / WAVE LENGTH
B) 0 = 45°
Fig.3.4.2. Lamb waves propagation in the unidirectional
graphite/epoxy composite plates
52
mode velocity has reduced significantly compared with Fig. 3.4.2.A
The Al mode velocity of a very thin ( hA - 0) plate approaches the
QL wave instead of the So mode velocity.
.
When 9 = 90°, Ao and So
modes in Fig. 3.4.2.0 show the same dispersion pattern as those for
an isotropic material as given in Fig. 3.4.3.
This reflects that the
wave propagation plane is transversely isotropic as expected.
As the thickness of the plate increases, the velocities of Ao
and So modes of every plot converge to the Rayleigh wave velocity.
As hA increases, Al and S1 mode velocities approach the QT wave
velocity.
When the plate is thick enough, the Rayleigh wave on
one surface can be regarded as a linear combination of Ao and So
modes, and vise versa in isotropic plates [73].
The same
interpretation can be applied to the composite plate.
The Ao
mode velocities are continuously increasing and reach 90% of the
Rayleigh wave velocity at h/2t, 0.5 regardless of the wave
propagation direction as shown in Fig. 3.4.4.
The So mode
velocities for a very thin plate ( hA < 0.4 ) are close, but show a large
rise very close to the fiber direction ( 0° 5 9 < 10 °) in Fig. 3.4.5.
53
3
SYMMETRIC MODE
-------
ANTISYM. MODE
2.5
SI
2
So
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
PLATE THICKNESS ( h ) / WAVE LENGTH
C) 0 = 900
Fig.3.4.2. Lamb waves propagation in the unidirectional
graphite/epoxy composite plates
PLATE THICKNESS ( h ) / WAVE LENGTH
Fig. 3.4.3. Lamb waves propagation in the aluminum plate.
54
PLATE THICKNESS ( h ) / WAVE LENGTH
Fig. 3.4.4. The Ao mode velocities by the change of wave
propagation directions in the unidirectional
graphite/epoxy composite plates.
10
0=0°
0000
8 =45°
8 =90°
3
2
---------- ------------ ------------------------------------------
1
oo
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
PLATE THICKNESS (h)/ WAVE LENGTH
Fig. 3.4.5. The S0 mode velocities by the change of wave
propagation directions in the unidirectional
graphite/epoxy composite plates.
55
CHAPTER 4
EXPERIMENTS
4.1.
Experimental Arrangement
Schematic diagrams of the experimental setup for single and
dual sensors are shown in Fig. 4.1.1 and Fig. 4.1.2, respectively.
Velocity measurements using these two methods will be compared
in section 4.4.
The experiments were conducted using a laser
ultrasonic wave generation system, wave detection sensors ( piezoelectric pin transducers ), and a high-speed digital data acquisition
system.
The ultrasonic wave generation system consists of a ruby laser
with its remote control unit.
The ruby laser produces a light pulse
with a wave length of 0.694 gm, a pulse duration of about 30 nsec, a
maximum energy of 1.2 J, and a beam diameter of about 11 mm.
It is
operated in a Q-switched mode in order to generate a strong single
impulse onto the specimen.
The remote control unit, which can
provide a trigger signal to the digital oscilloscope, is used to set the
laser power level and control the pockel cell operation.
The
digital oscilloscope can be triggered accurately by a PMT ( Photomultiplier tube ) as shown in Fig. 4.1.1.
This optical triggering
method can eliminate most significant triggering time errors except
for PMT risetime and the light flight time by the optical path
difference from the ruby laser to the PMT and a specimen surface.
For example, an optical path of 0.5 m produces only a 1.5 nsec time
56
ENERGY METER
\
SPECIMEN
FOCUSING
CYLINDRICAL LENS
BEAM
SPLITTER
Q - SWITCHED
RUBY LASER
7
REMOTE
CONTROL
PMT
UNIT
J
L
ULTRASONIC WAVE
GENERATION SYSTEM
WAVE DETECTION SENSOR
( PIN TRANSDUCER )
TRIGGER
SIGNAL
CH.1
1
PRINTER
PLOTTER
IBM PC
TEK 2342A
GURU SOFTWARE
DIGITAL
FFT PROGRAM
OSCILLOSCOPE
L
HIGH SPEED DIGITAL DATA ACQUISITION SYSTEM
Fig.4.1.1.
Schematic diagram of experimental setup
for the single pinducer technique.
57
ENERGY METER
FOCUSING
CYLINDRICAL LENS
/
Q-SWITCHED
RUBY LASER
SPECIMEN
BEAM
SPLITTER
r
1
REMOTE
CONTROL
UNIT
ULTRASONIC WAVE
GENERATION SYSTEM
WAVE DETECTION SENSORS
(DUAL PIN TRANSDUCERS)
TRIGGER
SIGNAL
CH.1
CH.2
r
PRINTER
PLOTTER
IBM PC
TEK 2342A
GURU SOFTWARE
DIGITAL
FFT PROGRAM
OSCILLOSCOPE
HIGH SPEED DIGITAL DATA ACQUISITION SYSTEM
Fig. 4.1.2.
Schematic diagram of experimental setup
for dual pinducer technique.
58
difference) [13].
On the other hand, the triggering time delay is
not important in Fig. 4.1.2 because only the time delay between dual
pinducers is measured.
Wave detection is provided by pinducers ( piezo-electric pin
transducers ) which have broad band response.
The outer
diameter of a pinducer is about 2.4 mm and the inner-core of the
piezoelectric material is 1.4 mm diameter.
Note that the pinducers
are used instead of a Michelson interferometer or a fiber optic
detection system which usually require a good reflective surface for
high signal-to-noise response.
Metal surfaces are often quite
reflective and can be polished easily.
On the other hand,
graphite/epoxy composites are poorly reflective and highly diffusive.
A thin mirror may be attached to the surface of the composite to
provide a reflective surface.
Preliminary studies show that when
a thin, small mirror is attached on a large mirror, the shape of a
Rayleigh wave on the thin mirror is the same as that on a large
mirror with a small additional time delay due to the attached mirror
thickness.
However, if a mirror is glued on a graphite/epoxy
composite, the transmitted Rayleigh wave shape on the attached
mirror is somewhat different from that on the composite surface.
Because of this, the piezoelectric pinducers have been chosen over
laser interferometry for use with graphite/epoxy composites.
The pinducer does require couplant as with conventional
piezoelectric transducers.
However, the small diameter of the
pinducer minimizes the possible variability of couplant thickness
59
and problems associated with surface roughness as long as the
pinducer is positioned perpendicular to the surface.
Tests show
that good signals can be obtained just by touching contact to the
surface.
Even an intentionally slight gap ( 0.1-0.3 mm ) between the
pinducer and the surface results in only a small attenuation of wave
signals where couplant is used.
With the single pinducer approach, wave propagation time can
be measured from the PMT trigger signal in channel 2 to the slight
initial rise of the Rayleigh wave as shown in Fig. 4.1.3.
When the
dual pinducer technique is used, direct measurements of pinducer
distance and time interval between peak to peak signals are
straightforward with each signal stored in channel 1 and 2. The
time interval can be directly measured by using a curser on the
digital oscilloscope.
It is possible that the Rayleigh wave can be
affected by the presence of the pinducer and couplant.
This
contact problem can be minimized by using a laser induced line
source and just the touching contact to the surface without further
pressure.
A train of Rayleigh waves induced by a line source
produces a stronger signal than a point source and should minimize
possible perturbations produced by the front pinducer. A graphite/
epoxy composite is very attenuative and attenuation from the same
source is expected to be measured accurately by the dual pinducer
technique.
The high speed digital data acquisition system consists of a
digital oscilloscope ( Tektronix 2342A ) which can store 1024 digital
60
Wave propagation time
Channel 1
Channel 2
(PMT trigger)
A) EXPERIMENT WITH A PIN TRANSDUCER
Channel 1
(PIN #1)
Channel 2
---,le-A-1
(PIN #2)
Wave propagation time
B) EXPERIMENT WITH DUAL PIN TRANSDUCERS
Fig. 4.1.3. Measurements of Rayleigh wave propagation time
by the single and the dual pinducer techniques.
61
data points, and an IBM computer with digital interface software
( GURU ) and an FFT ( Fast Fourier Transform ) spectrum analysis
program.
4.2. Preparation of Specimen
Four unidirectional graphite/epoxy composite plates with 60%
fiber volume fraction were cured and fabricated by Wyle Labs at
Edwards Air Force Base as given in the following table:
Table 4.2.1. Dimensions of graphite/epoxy plates ( T300/5208 ).
Plate No.
Length (inch)
Width (inch)
1
6
6
0.034
(0.87 mm)
2
6
6
0.067
(1.7 mm)
3
6
6
0.128
(3.25 mm)
4
6
6
0.6395
(16.25 mm)
Thickness (inch)
4.3. Experimental Procedure
The experimental setups were essentially the same for
Rayleigh wave and Lamb wave measurements.
Plate no. 4 was
very thick compared to the others and was regarded as approaching
a semi-infinite specimen relative to the induced surface wave length.
Therefore, the velocity and attenuation measurements of Rayleigh
62
waves were conducted on plate no. 4.
The other three plates were
thin enough to use for Lamb wave measurements.
One edge of
each plate was held tight by a table vise and other edges were free of
constraints.
Surfaces of plates were free of stress.
Any low
frequency, flexural plate vibrations initiated by the laser deposition
would have much slower velocities than the induced ultrasonic
waves.
Each measurement was completed before the arrival of any
flexural vibration of the plate.
Therefore, the one-edge-fixture
seems to be convenient and sufficient for this experiment.
As shown in Fig. 4.3.1, the plate was rotated with 10 degree
increments for each laser deposition test.
Line sources
by focusing laser beam
Pinducer
distance = 20 60 mm
Spacing = 10 - 20 mm
76 mm
/
0
50°
10 increments
47 mm
60 mm
Fiber direction
Fig. 4.3.1. Layout of laser deposition tests for Rayleigh
and Lamb waves generation ( 150 mm by 150 mm ).
63
Test Procedure (refer to Fig. 4.1.1. and Fig. 4.1.2.)
1) The ruby laser was set to generate maximum energy
resulting in maximum wave amplitude.
Because the
S/N ratio ( Signal-to-Noise ratio ) was very low in the
thermoelastic regime, every experiment was conducted in
the ablation regime.
2) An energy meter, set perpendicular to the laser beam path,
received about 17 % of the laser energy as reflected by a
plane glass plate placed 45° at to the laser beam path.
Most experiments were monitored by an energy meter to
compare data at same energy level.
3) The laser beam was focused to a line of about 0.5 mm width
and 15 mm length using a cylindrical lens.
The focusing
width controls the power density of the laser beam and the
corresponding induced wave amplitude, but does not
influence the measurement accuracy of the wave
propagation time.
4) The pinducers were positioned perpendicular to and just
touching the specimen surface without further pressure.
Each pinducer was held in position by a bracket attached to
a strong magnetic base.
The distance from the line
source to a pinducer center or the spacing distance of
the dual pinducers was measured accurately using a
micrometer.
64
5) At least 3 minutes of shot interval time was used between
laser firing to maintain stability of output.
6) The digital oscilloscope was triggered by a signal from the
remote control unit of the laser power supply or from the
PMT output.
The first arrival of the wave was recorded
on channel 1 and the second arrival or the PMT signal on
channel 2 of the digital oscilloscope.
Wave propagation time was measured accurately on the
oscilloscope screen by using the vertical time cursers and
wave amplitudes by using the horizontal voltage cursers.
The corresponding wave propagation velocity and
attenuation coefficient were then calculated.
7) The digital signal was transferred from the oscilloscope to
an IBM PC computer using GURU software.
FFT spectrum
analysis and digital filtering were then performed.
Hard
copies of the wave signals were made on a laser printer.
4.4. Experimental Results and Discussion
4.4.1. Rayleigh Waves
As discussed in section 4.1, dual pinducers were expected to
be accurate for wave velocity and attenuation measurements because
many measurements could be made at each single experiment with
the same source.
Before conducting the main experiments, the
characteristics of pinducers were studied.
65
A pinducer was located at 20 mm distance from a line source
and a wave signal was obtained.
Then, the pinducer was moved
10 mm farther from the line source and another wave signal was
recorded and so on.
This test was performed at the same energy
level for wave propagation parallel and perpendicular to the fiber
directions as shown in Fig. 4.4.1 and Fig. 4.4.2, respectively.
For
comparison purposes, time scale and voltage output scale are the
same for every waveform.
Each trace shows a disturbance at time
zero caused by electrical noise from the ruby laser at the time of
firing.
The largest sharp signal of each trace is a Rayleigh wave.
Each Rayleigh wave arrival time is found to be proportional to the
corresponding distance from the line source.
Thus, the Rayleigh
wave velocity is independent of wave propagation distance.
The
wave is also non-dispersive maintaining approximately the same
wave length and shape.
Wave amplitude is found to decrease
with propagation distance due to the high attenuation of the
The wave amplitude of the first large
graphite/epoxy composite.
negative signal was used for attenuation measurements.
The relation between the laser energy and the pinducer
output was found to be linear as shown in Fig. 4.4.3.
A pinducer
was fixed at 20 mm distance from a line source and the pinducer
output voltage of the first large negative peak was measured as the
laser energy was changed.
Both correlation coefficients for 9 = 0°
and 0 = 90° show high linearity.
The wave amplitude may be
attenuated somewhat by contact with the pinducer, but the output
66
70
60
50
40
30
20
10
5
0
10
15
20
25
30
35
TIME (micro second)
Fig. 4.4.1. Rayleigh waves in the unidirectional graphite/
epoxy composite as a pinducer moves from
a line source ( 0 = 0° ).
60
---
------77-1
50
..-""
40
30
20
10
10
5
0
5
10
15
20
25
30
35
TIME (micro second)
Fig. 4.4.2. Rayleigh waves in the unidirectional graphite/
epoxy composite as a pinducer moves from
a line source ( 0 = 90° ).
40
67
40
Linear regression ( 90 degree )
35
Y = 0.43606 + 0.11995 X
Correlation coeff. = 0.99535
30
25
20
15
10
Linear regression ( 0 degree )
Y = 0.38097 + 0.05918 X
Correlation coeff. = 0.99533
5
Oo
50
100
150
200
250
300
350
LASER ENERGY (mJ)
Fig. 4.4.3. Relation between the laser energy and
the pinducer output.
400
68
voltages of piezo-electric crystals should be linear with small
amplitudes if the attenuation of wave amplitude by a pinducer is
negligible.
Also, if there was any high nonlinearity among the
laser energy, the wave amplitude, and the pinducer output, these
high correlation coefficients would not be obtained.
Therefore,
Rayleigh wave amplitudes and laser energies show good linearity at
lower laser energy levels.
At higher energy levels, this may not
be true.
Tests using the dual pinducer technique resulted in many
problems.
First, the outputs of each pinducer were different
from one another at the same laser energy even though each one
had a high output linearity and showed repetitive signals.
The
wave amplitude after passing a pinducer was attenuated about 10%
more compared to that without contact.
This attenuation
amount seemed to be a function of many factors, i.e., the
characteristics of each pinducer, wave propagation directions,
distances from a line source, etc.
Also, minor dispersion
occurs so that the measurement of wave propagation time was
different when wave shape was changing.
The attenuation coefficient Y is defined as follows [33]:
20 logi 0 I
V2
,
I
Ax
where V1 = wave amplitude at a reference point x1
V2 = attenuated wave amplitude at point x2
(4.4.1)
69
Ax = x2 - x1 = distance between the two point of
measurements.
The measurements of wave amplitudes at the first major negative
going signal for 0 = 0° and 0 = 90° were plotted with each 10 mm
interval as shown in Fig. 4.4.4.
Constant attenuation curves were
calculated using the first and the last pinducer output.
Table 4.4.1. Attenuation calculation of
a graphite/epoxy composite
angle
(0)
interval
attenuation
average
(mm)
(dB/cm)
attenuation
20 - 30
0° 0
90°
30 40
40 50
50 60
-1.859
-2.948
-2.923
-3.757
20 30
30 40
40 50
-6.508
-4.358
-1.987
-2.872 dB/cm
-4.284 dB/cm
Along the fiber direction, experimental data were close to the
calculation and the attenuation coefficient was approximately
constant. However, at 0 = 90°, the Rayleigh wave was attenuated
rapidly near the line source and more slowly far from the line
source.
Wave attenuation appears to change with the distance
from the line source.
If composites are damaged by impact, then
wave propagation velocity is expected to decrease and attenuation to
70
35
0000
30
Experiment ( 0 degree ) Experiment ( 90 degree )
Constant Attenuation
( 0 degree )
Constant Attenuation
( 90 degree )
XXXX
25
---
20
15
10
5
Oo
10
20
30
40
50
60
70
DISTANCE FROM A LINE SOURCE (mm)
Fig. 4.4.4. Attenuation of Rayleigh wave amplitudes
in the graphite/epoxy composite
when 0 = 0° & 90° ( laser energy = 205 ml ).
80
71
increase.
Therefore, attenuation measurements along the fiber
direction would appear to be more useful for NDT of composite
damage by impact.
Some plots of experimental Rayleigh wave forms and FFT
analysis are given in Appendix D.
Each plot shows a similar wave
form with Rayleigh wave having a frequency of about 0.8 ± 0.05 MHz.
The calculated wave length of the Rayleigh wave is about 2.5 mm
when 0 = 0° and 1.8 mm when 0 = 90°.
Therefore, plate no. 4 is
about 6.5 - 9.0 wave lengths thick and the simulation of a semi-
infinite composite as discussed in section 4.3. should be valid.
The calculations and experimental results for Rayleigh wave
phase velocities are plotted as shown in Fig. 4.4.5.
Experimental
data were obtained by the single pinducer technique and show a
very good agreement with calculated values.
Rose and Pilarsky
formulated the same equations as given in section 3.1. to calculate
the Rayleigh wave velocity of unidirectional graphite/epoxy
composites ( 60% fiber volume fraction ) [57].
Their results
appears to be in error due to incorrect eigen value and eigen vector
calculations caused by the direct solution of bicubic equations as
discussed in section 3.2.1.
They used conventional transducers
with frequencies of 1 MHz and 2 MHz, and generated Rayleigh waves
along one surface line and received at two others separated
10 mm.
Such a small separation was chosen not only due to large attenuation, but mainly to avoid any difficulties with the interpretation of
72
2.5
experiment
calculation
experiment by Rose
calculation by Rose
2
1.5
1
0.5
0
0
0.5
1
1.5
2
25
VELOCITY (KM/SEC), PARALLEL TO FIBER
Fig. 4.4.5. Calculations and experiments of Rayleigh wave
phase velocities in the unidirectional graphite/
epoxy composite and comparison with Rose's
results [see reference 57].
73
the received signals, due to finite thickness of the composite layer
and possible reflected waves from the far side.
As shown in Fig. 4.4.5, the velocity calculations by Rose et al
[57] were very close to calculations of this work along the fiber
direction (0 = 0 °) because an eigen value and an eigen vector could
be found correctly without the coupling difficulty.
difference is probably due to material constants used.
The small
Rose's
calculations show increasing deviations compared to present results
with increasing values of 0, and even show a sharp change between
0 = 85° and 90° where material properties are nearly the same.
Rose's experimental results can not be compared directly with
their calculations because material constants of the unidirectional
graphite/epoxy composite were unknown.
However, their
experimental results also showed significant change near 0 = 90°
similar to their calculations.
Such a rapid change does not appear
reasonable without significant changes of material constants which
do not occur when approaching 0 = 90° .
The same experiments were repeated using dual pinducers
and showed poor agreement with calculations, Fig. 4.4.6. As pointed
out in section 4.4.1, dual pinducers caused additional attenuation
and dispersion.
Also, the short spacing of dual pinducers was
necessary because of the high attenuation, and even small changes of
spacing could produce the rather large errors.
74
2.5
oooo
xxxx
experiment with a pinducer
experiment with dual pinducers
2
Rayleigh wave calculation
1.5
x
x
x
1
x
0.5
0
0
0.5
1
1.5
2
25
VELOCITY (KM/SEC), PARALLEL TO FIBER
Fig. 4.4.6. Comparison between the single and the dual
pinducer techniques in Rayleigh wave velocity
measurements.
75
4.4.2. Lamb Waves
Before considering Lamb waves in graphite/epoxy composites,
Lamb waves in aluminum were briefly considered for a case study.
The laser beam was focused to a line source on an aluminum plate of
0.8 mm thickness and waves were detected at 35 mm distance from
the line source.
As shown in Fig. 4.4.7.A, the So wave of 2.2 MHz
arrives at 6.3 gsec followed by a surface wave with sharp large
amplitudes and a frequency of 1.95 MHz.
Then, the Ao wave of
lower frequency ( 0.8 MHz) arrives at about 10 ilsec with the S1 wave
of higher frequency ( 4.4 MHz ).
As shown in Fig. 4.4.7.B, the peak
frequency of each wave mode can be clearly found in aluminum.
The waveforms and corresponding FFT analysis for the
thinnest plate, no. 1, are given in Appendix E.
Note the presence
of a surface wave in the plate wave propagation.
If the laser
energy is low enough, the surface wave will not appear, but the
signal-to-noise ratio will be very low.
The surface wave has a
frequency of about 0.77 ± 0.04 MHz which is nearly the same as that of
the Rayleigh wave, and its wave length is about 2.7 mm when 0 = 0°
and 2.0 mm when 0 = 90°.
The wave length of the plate surface
wave is two or three times larger than the thickness 0.87 mm of plate
no. 1.
Therefore, the "plate surface wave" should not be called a
"Rayleigh wave" which is a surface wave in a semi-infinite medium.
As an example of thin plate wave analysis, Lamb wave propaga-
tion in plate no. 1 when 0 = 90° is considered.
A pinducer was
76
100
50
0
0
E-4
o
-50
-100
a.
-150
-10
-5
0
5
10
20
15
25
30
35
40
TIME ( micro second )
Fig. 4.4.7.A Laser generated ultrasonic waves in an aluminum
plate of 0.8 mm thickness.
x106
3
Surface wave
2.5
2
AO
1.5
SO
1
0.5
2
3
4
5
6
FREQUENCY ( MHz )
Fig. 4.4.7.B
Frequency analysis of Fig. 4.4.7.A.
7
77
located 20 mm from a line source.
As shown in Fig. 4.4.8.A and B,
the So wave arrives at 7 lisec followed by a surface wave with a sharp
large amplitude and a frequency of 1.74 MHz.
Then, the Ao wave of
lower frequency ( 0.25 MHz ) arrives with the S1 wave of higher
frequency ( 2.0 MHz) superposed .
To find an accurate arrival of
each mode, digital filtering was performed as shown in Fig. 4.4.9.
In spite of the low pass digital filtering ( cut-off frequency
0.7 MHz
a.--.
), the true Ao wave is difficult to find because of the "ringing" sideeffect of the filtered signal.
The band pass digital filtering
between 0.7 and 1.7 MHz shows the surface wave.
The high pass
digital filtering ( cut-off frequency E- 1.7 MHz) shows the arrival of S1
wave at about 7 ilsec before the surface wave arrival.
is, however, difficult to find by filtering.
The So wave
In general, the exact
arrivals of the laser generated Lamb waves are difficult to find, and
frequently the Ao, So, S1 Lamb waves, and the surface wave are
superposed on each other.
On the other hand, the "leaky Lamb
wave technique" by conventional transducers can generate Lamb
waves without the surface wave and can find each Lamb wave mode
more easily.
Therefore, this experiment was not continued
further for plates no. 2 and 3.
The evaluation of Lamb wave
velocities using laser generated Lamb waves is not recommended.
In Fig. 4.4.10, the surface wave phase velocities of plate no. 1
are plotted against Rayleigh wave phase velocities of plate no. 4.
The plate surface waves when 9 E 0° are about 7 % faster than
Rayleigh waves, but difference is much less when 0 -E- 90°.
This
78
80
surface wave
60
40
e-a
20
a.
0
0
z
-20
Si
So
40
5
0
5
15
10
20
25
35
30
40
TIME ( micro second )
Fig. 4.4.8.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
perpendicular to the fiber direction.
x105
14
Surface wave
12
S1
10
AO
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. 4.4.813
Frequency analysis of Fig. 4.4.8.A.
7
79
100
50
0
ers
50
a) 100
a)
4. -150
0
200
:2' 250
A
300
350
15
10
5
0
5
10
15
20
25
30
TIME ( micro second )
Fig. 4.4.9.
Digital filtering analysis for Lamb wave signal of
Fig. 4.4.8.A
i) Low pass filtering with a cut-off frequency
of 0.7 MHz,
fi) Band pass filtering between 0.7 and 1.7 MHz,
iii) High pass filtering with a cut-off frequency of
1.7 MHz.
80
experiment ( h = 16.25 mm )
experiment ( h = 0.87 mm )
Rayleigh wave calculation
0
0.5
1
1.5
2
25
VELOCITY (KM/SEC), PARALLEL TO FIBER
Fig. 4.4.10. Polar plot of surface wave phase velocities
on the graphite/epoxy composite plates of
thickness 0.87 & 16.25 mm.
81
increase of the plate surface velocity when 9 a 0°, rather than when
9 a 90°, may be useful for the detection of delamination in thick
composites because the delaminated part of the composite can be
represented as a thin plate with traction-free surfaces.
82
CHAPTER. 5
CONCLUSIONS AND RECOMMENDATIONS
The results of the laser generated Rayleigh and Lamb waves in
graphite/epoxy composites can be summarized as followings:
1) The Rayleigh wave phase velocities can be calculated
accurately by a new eigenvalue search method.
2) The relation between the laser energy and the pinducer
output was found to be linear.
Therefore, the relation of
the Rayleigh wave amplitude and laser energy should be
approximately linear at least in lower laser energy levels.
3) Attenuation along the fiber direction shows an approximate
constant value and could provide means for NDT of
composite damage by impact.
4) The dual pinducers technique suffered from the additional
attenuation and dispersion by the front pinducer.
This
approach is believed to be less accurate than the single
pinducer technique for velocity and attenuation
measurement.
5) Laser generated Rayleigh waves with detection by a
pinducer shows better accuracy agreement with theory
than the conventional transducer technique for Rayleigh
wave velocity measurement.
83
6) Laser generated Lamb waves with detection by a pinducer
lead to difficulties in finding precise arrivals of each Lamb
wave mode.
7) The increase of the plate surface velocity when 0 a 0°, rather
than 0 -2 :- 90°, may be useful for the detection of delamination
because the delaminated part of the composite can be
simulated as a thin plate with traction-free surfaces.
The recommendations for future study include:
1) The measurement of Rayleigh wave attenuation in
unidirectional graphite/epoxy composites with impact
damage.
2) The detection of surface wave velocity changes in
unidirectional graphite/epoxy composites with simulated
delaminations.
3) The calculation of Rayleigh wave velocity in multi-layer,
general graphite/epoxy composites.
4) The impact transient solution of displacement and velocity
of the Rayleigh waves in unidirectional graphite/epoxy
composites using a green's function.
84
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"Laser-based ultrasonics on gr/epoxy composite: interferometer
detection"
Review of Progress in Quantitative NDE, Plenum Press,
NewYork, NY., Vol. 9A, 1990, pp. 479-485.
71. Tsai, S. W.
"Composites Design"
Think composite, 4th ed., 1988.
72. Vary, A. and K. J. Bowles
"Ultrasonic Evaluation of the Strength of Unidirectional
Graphite-Polymide Composites"
NASA TM-X-73646, Apr. 1977.
73. Viktorov, I. A.
"Rayleigh and Lamb waves"
Plenum Press, New York, 1967, pp. 1-122.
74. Wetsel, G. C.
"Photothermal generation of thermoelastic waves in composite
media"
IEEE Trans. UFFC, 1986, UFFC-33(5), pp. 450-461.
92
75. White, R. M.
"Elastic wave generation by electron bombardment of
electromagnetic wave absorption"
Journal of Applied Physics, Vol. 34, 1963, pp. 2123-2124.
76. White, R.M.
"Generation of elastic waves by transient surface heating"
Journal of Applied Physics, Vol. 34(12), 1963, pp. 3559-3567.
77. Worlton, D. C. "Experimental Confirmation of Lamb Waves at
Megacycle Frequencies"
Journal of applied physics, Vol.32, 1961, pp. 967-971.
78. Worm ley, S. J. and R. B. Thompson
"A Semi-Automatic System for the Ultrasonic Measurement of
Texture"
Review of Progress in Quantitative, NDE Plenum Press,
New York, NY., Vol. 6A, 1987, pp. 951-956.
APPENDICES
93
APPENDIX A
PROGRAM FOR RAYLEIGH WAVES
% *************************************************************
% CSW.M for Rayleigh wave velocity calculation.
%
%
This program finds the accurate Rayleigh wave velocity which can satisfy
the zero value of BCD(boundary condition determinant) within tolerance.
% The proper search interval can be found by PD.M.
% If there is no solution, search interval will move out of the
%
initial interval by the modified golden section method.
% VV
= monitoring velocity according to E(1)
= smaller velocity in search interval
v2
= larger velocity in search interval
E(i)
= eigen value according to the assumed velocity
F(i)
= BCD corresponding to E(i)
= stiffness matrix of plate
% AA
= all or part of A matrix components
% THETA = wave propagation direction relative to fiber direction (degree)
% RHO = density of plate
% TOL
= tolerance for iteration termination
% ARM = arm for search interval change
%
Ll
= X direction cosine of wave propagation vector
%
L2
= Y direction cosine of wave propagation vector
% X (L3) = 3 roots of L3 according to the assumed velocity
%
A(:,i)
= eigen vector components corresponding to X(i)
% *************************************************************
clear
format short e
i = sqrt(-1)
getdata
%
%
%
%
vl
%C
vl =input(' vl initial =')
v2 =input(' v2 final =')
E(1) = RHO*vl*v1
E(4) = RHO*v2*v2
F(1) = Eval(E(1),C13,C23,C33,C44,C55,AA,THETA) ;
F(4) = Eval(E(4),C13,C23,C33,C44,C55,AA,THETA) ;
index = 1
TOL = le-5
count = 0
while ((E(4)-E(1))/E(4)) > TOL
ARM = (E(4) - E(1))*0.618
E(2) = E(4) ARM
E(3) = E(1) + ARM
if (index=3), F(2) = Eval(E(2),C13,C23,C33,C44,C55,AA,THETA);, end
if (index=2), F(3) = Eval(E(3),C13,C23,C33,C44,C55,AA,THETA);, end
[MIN,index] = min(F)
94
if(index==1)
E(4) = E(2)
;
F(4) = F(2)
;
E(1) = E(1) - ARM; F(1) = Eval(E(1),C13,C23,C33,C44,C55,AA,THETA) ;
elseif(index==2)
E(4) = E(3)
E(3) = E(2)
elseif(index==3)
E(1) = E(2)
E(2) = E(3)
else
;
F(4) = F(3)
F(3) = F(2)
;
F(1) = F(2)
F(2) = F(3)
;
E(4) = E(4) + ARM; F(4) = Eval( E( 4 ),C13,C23,C33,C44,C55,AA,THETA) ;
E(1) = E(3)
;
F(1) = F(3)
,
end
VV = sqrt(E(1)/RHO)
count = count +1
;
[count real(VV) E(1) E(4) F(1) F(4) ]
end
% end while
A(3,3) = 0
;
[X(1) A(:,1)] = eigen(1,E(1),C33,C44,C55,AA)
[X(2) A(:,2)] = eigen(2,E(1),C33,C44,C55,AA)
[X(3) A(:,3)] = eigen(3,E(1),C33,C44,C55,AA)
if THETA ==O
temp = X(3)
X(3) = X(2)
X(2) = temp
tempi = A(:,3)
A(:,3) = A(:,2)
A(:,2) = temp 1
end
S = MatrixB(C13,C23,C33,C44,C55,A,L1,L2,X) ;
VV = sqrt(E(1)/RHO)
% save csw VV S A X Ll L2 C11 C12 C13 C22 C23 C33
95
% *************************************************************
% Getdata.m
%
% This M-file is to get the material constants of plate
and calculates components of stiffness matrix.
%
% RHO
= Density
% P21, P23 = Poisson's ratio
% El, E2
% G12
= Young's modulus
= Shear modulus
(following Tsai's convention)
% pi = 3.141592
% *************************************************************
THETA = input('Enter angle = ')
ANGLE = THETA*pi/180
RHO = 1.52e-6
P21 =0.31
P23 = 0.54
El = 138.5e-6
E2 = 11.e-6
G12 = 6.3e-6
P12 = P21*E2/E1
D = 1/(1+P23)/(1-P23-2*P21*P12)
C11 = (1-P23*P23)*D*E1
C22 = (1-P21*P12)*D*E2
C33 = C22
C12 = P21*(1.+P23)*D*E2
C13 =C12
C23 = (P23+P21*P12)*D*E2
C44 = (1.-P23-2*P21*P12)*D*E2/2
C55 = G12
C66 = G12
Ll = cos(ANGLE)
L2 = sin(ANGLE)
AA(i) = C11 *Ll*L1 +C66*L2*L2
AA(2) = C66*Ll*L1 +C22*L2*L2
AA(3) = C55*L1 *L1 +C44*L2*L2
AA(4) = (C12+C66)*Ll*L2
AA(5) = (C13+C55)*L1
AA(6) = (C23+C44)*L2
; % Kg/mm/mm/mm
,
,
; % Kg/mm/microsec/microsec
; % These units make a velocity
; % units in mm/microsec.
96
% *************************************************************
% Eval.m for csw.m
% This M-file calculates the value of boundary condition determinant
% and returns absolute value of it.
% *************************************************************
function y = Eval(E,C13,C23, C33, C44,C55,AA,THETA)
Ll = cos(THETA*pi/180)
L2 = sin(THETA*pi/180)
A(3,3) = 0
[X(1) A(:,1)] = eigen(1,E,C33,C44,C55,AA)
[X(2) A(:,2)] = eigen(2,E,C33,C44,C55,AA)
[X(3) A(:,3)] = eigen(3,E,C33,C44,C55,AA)
if THETA ==O
temp = X(3)
X(3) = X(2)
X(2) = temp ;
temp 1 = A(:,3)
A(:,3) = A(:,2)
A(:,2) = temp 1
end
S = MatrixB (C13, C23, C33, C44,C55,A,L1 ,L2,X)
y = abs(det(S))
% *************************************************************
% Eigen.m
% This M-file is to find imaginary root of L3 and calculate
% corresponding eigen value and eigen vector.
% *************************************************************
function [Y,A] = eigen(K,E,C33,C44,C55,AA)
i = sqrt(-1)
LOW =
FL = Avalue(LOW,K,E,C33,C44,C55,AA)
while FL > 0
LOW = LOW - i
FL = Avalue(LOW,K,E,C33,C44,C55,AA)
end
HIGH = 0
FH = Avalue(HIGH,K,E,C33,C44,C55,AA)
index = 1
TOL = le-10
while abs(HIGH-LOW) > TOL
MID = (LOW+HIGH)/2
97
FM = Avalue(MID,K,E,C33,C44,C55,AA)
if FM > 0
HIGH = MID
= FM
else
LOW = MID
FL = FM
end
end
T = MatrixA(LOW,C33,C44,C55,AA)
[VECTOR VALUE] = eig(T)
Y = LOW
A = VECTOR(:,K)
% *************************************************************
% Avalue.m
% This M-file returns difference between the assumed eigen value and
% the eigen value corresponding to X.
% When this difference approaches to zero, root of L3 is found.
% *************************************************************
function y = Avalue(X,K,E,C33,C44,C55,AA)
Z = MatrixA(X,C33,C44,C55,AA)
EIGVALUE = eig(Z)
y = EIGVALUE(K)-E
% *************************************************************
% MatrixA.m
% This M-file is for component calculation of A matrix.
% *************************************************************
function y=MatrixA(X,C33,C44,C55,AA)
A(1,1) = AA(1) +C55*X*X
A(2,2) = AA(2) +C44*X*X
A(3,3) = AA(3) +C33*X*X
A(1,2) = AA(4)
A(2,1) = A(1,2)
A(1,3) = AA(5)*X
A(3,1) = A(1,3)
A(2,3) = AA(6)*X
A(3,2) = A(2,3)
y=A
98
% *************************************************************
% MatrixB.m
% This M-file is to calculate components of boundary condition matrix (B matrix).
% *************************************************************
function y= MatrixB(C13,C23,C33,C44,C55,A,L1,L2,X)
B(1,1) = C55*(A(1,1)*X(1) +A(3,1)*L1)
B(1,2) = C55*(A(1,2)*X(2) +A(3,2)*L1)
B(1,3) = C55*(A(1,3)*X(3) +A(3,3)*L1)
B(2,1) = C44*(A(2,1)*X(1) +A(3,1)*L2)
B(2,2) = C44*(A(2,2)*X(2) +A(3,2)*L2)
B(2,3) = C44 *(A(2,3) *X(3) +A(3,3)*L2)
B(3,1) = C13*A(1,1)*L1 +C23*A(2,1)*L2 +C33*A(3,1)*X(1) ;
B(3,2) = C13*A(1,2)*L1 +C23*A(2,2)*L2 +C33*A(3,2)*X(2) ;
B(3,3) = C13*A(1,3)*L1 +C23*A(2,3)*L2 +C33*A(373)*X(3) ;
y=B
% *************************************************************
% CD.M
% This M-file is to calculate the value of BCD( boundary condition determinant)
% within an interval selected and show the pattern of BCD.
% After finding a small interval including root,
% run csw.m to find accurate root value.
% *************************************************************
clear
format short e
i = sqrt(-1)
getdata
A(373) = 0
V1 = input(' Initial = ')
V2 = input(' Increment = ')
V3 = input(' Final = ')
index = 1
for V = V1 :V2:V3
E = RHO*V*V
[X(1) A(:,1)] = eigen(1,E,C33,C447C55,AA)
[X(2) A(:,2)] = eigen(2,E,C33,C44,C55,AA)
[X(3) A(:,3)] = eigen(3,E,C33,C44,C55,AA)
if THETA ==O
temp = X(3)
X(3) = X(2)
X(2) = temp
tempi = A(:,3)
A(:,3) = A(:,2)
99
A(:,2) = tempi
end
S = MatrixB(C13,C23,C33,C44,C55,A,L1,L2,X)
V = sqrt(E/RHO)
detl(index) = real(det(S))
det2(index) = imag(det(S))
[V detl(index) det2(index) ]
index = index +1
end
V = (V1:V2:V3)'
out=[V detl' det2' ]
plot(V, detl',':',V,det2',' -')
title(' DETERMINANT VALUE OF S MATRIX ')
text(0.5,0.85,'
Real value of det(S) ','sc')
text(0.5,0.8,'
Imag. value of det(S) ','sc')
xlabel(' VELOCITY ( MM / MICRO SEC) ')
ylabel(' DETERMINANT VALUE')
% *************************************************************
% CSWD.M
% This M-file is to calculate and plot displacements of Rayleigh wave.
% *************************************************************
clear
load csw45
CC(1) = 1
CC(2) = (S(2,1)*S(1,3)-S(1,1)*S(2,3)) /(S(1,2)*S(2,3)-S(1,3)*S(2,2))
CC(3) = -(S(3,1)+S(3,2)*CC(2))/S(3,3)
i = sqrt(-1) ;
XX = 0
YY = 0
WL = 1
;
K = 2*pi/WL
TT = 0
N=1
;
fl = 50
f2 = 5
for ZZ = 0:-WL/f1:42*WL
DX(N) = 0 ;
DY(N) = 0
;
DZ(N) = 0 ;
for J = 1:3
DX(N) = DX(N) +CC(J)*A(1,J)*exp(i*K*(L1*XX +L2*YY +X(J)*ZZ -VV*TT));
DY(N) = DY(N) +CC(J)*A(2,J)*exp(i*K*(Ll*XX +L2*YY +X(J)*ZZ -VV*TT));
DZ(N) = DZ(N) +CC(J)*A(3,J)*exp(i*K*(L1*XX +L2*YY +X(J)*ZZ -VV*TT));
end
N=N+1
end
[DX(1) DY(1) ]
ZZ = (0:-WL/f1:42*WL)'
disp = [DX',DY',conj(DZ')]
100
DZ = DZ*(-i)
DX = DX/DZ(1)
DY = DY/DZ(1)
DZ = DZ/DZ(1)
,
;
% Phase change
% Normalization
;
,
plot(ZZ, DX',':',ZZ,DY','--',ZZ,DZ','-')
title(' X,Y,Z DISPLACEMENT PLOT : THETA = ? ')
text(0.2,0.85,'
X disp. ','sc')
text(0.2,0.8, '- Y disp. ','sc')
text(0.2,0.75;
Z disp.','sc')
xlabel(' Z / WAVE LENGTH')
ylabel(' NORMALIZED DISPLACEMENT')
101
APPENDIX B
PROGRAM FOR LAMB WAVES
% *************************************************************
% PP.M for plate problem
%
%
%
%
%
This program finds the accurate Lamb wave velocity which can satisfy
the zero value of BCD(boundary condition determinant) within tolerance.
The proper search interval can be found by PD.M.
If there is no solution, search interval will move out of the
initial interval by the modified golden section method.
%V
% V1
% V2
%
%
E(i)
F(i)
%C
= monitoring velocity according to E(1)
= smaller velocity in search interval
= larger velocity in search interval
= eigen value according to the assumed velocity
= BCD corresponding to E(i)
= stiffness matrix of plate
= all or part of A matrix components
= wave number K * plate thickness H
% AA
% KH
% THETA = wave propagation direction relative to fiber direction (degree)
% RHO = density of plate
% TOL
= tolerance for iteration termination
% ARM
= arm for search interval change
= X direction cosine of wave propagation vector
L2
= Y direction cosine of wave propagation vector
% X (L3) = 6 roots of L3 according to the assumed velocity
% A(:,i)
= eigen vector components corresponding to X(i)
% *************************************************************
%
%
Ll
% --- Initialize and get material constants.
clear
; format short e ; i = sqrt(-1)
getconst ;
% Call Getconst.m
V1 = input(' Beginning of interval = ') ; V2 = input(' End of interval = ')
E(1) = RHO*V1*V1
;
E(4) = RHO*V2*V2
F(1) = DETvalue(E(1),C,AA,KH,THETA) ;
F(4) = DETvalue(E(4),C,AA,KH,THETA) ;
index = 1 ; TOL = le-5 ; count = 0
;
%---- Start search until tolerance is met.
% This is the modified golden section method.
while ((E(4)- E(i))/E(4)) > TO
ARM = (E(4) E(1))*0.618 ; E(2) = E(4) - ARM ; E(3) = E(1) + ARM ;
if (index=3), F(2) = DETvalue(E(2),C,AA,KH,THETA);, end
if (index ' =2), F(3) = DETvalue(E(3),C,AA,KH,THETA);, end
[MIN,index] = min(F)
102
%---- Future search interval depends on minimum of F.
%
(Modified golden section method)
if(index==1)
Minimum is toward the left of interval, so move interval to the left.
E(4) = E(2) ; E(1) = E(1) - ARM
F(4) = F(2) ; F(1) = DETvalue(E(1),C,AA,KH,THETA) ;
elseif(index==2)
Minimum is close to E(2), shrink interval to E(2).
E(4) = E(3) ; E(3) = E(2) ; F(4) = F(3) ; F(3) = F(2) ;
elseif(index==3)
Minimum is close to E(3), shrink interval to E(3).
E(1) = E(2) ; E(2) = E(3) ; F(1) = F(2) ; F(2) = F(3) ;
else
Minimum is toward the right of interval, so move interval to the right.
E(4) = E(4) + ARM
; E(1) = E(3)
;
F(4) = DETvalue(E(4),C,AA,KH,THETA) ; F(1) = F(3)
;
end
% end if
V = sqrt(E(1)/RHO) ; count = count +1
[count real(V) E(1) E(4) F(1) F(4) ]
end
% end while
%---- Final result calculation by E(1).
A(3,6) = 0 ; E = E(1) ; e = sort(eig(Amatrix(0,C,AA))) ;
if(E <= e(1))
[X(1) A(:,1)] = ieigen(1,1,E,C,AA) ;
[X(3) A(:,3)] = ieigen(2,1,E,C,AA) ;
[X(5) A(:,5)] = ieigen(3,1,E,C,AA) ;
X(2) = -X(1) ; A(:,2) = conj(A(:,1)) ;
X(4) = -X(3) ; A(:,4) = conj(A(:,3)) ;
X(6) = -X(5) ; A(:,6) = conj(A(:,5)) ;
elseif(e(1)<E & E<(2))
[X(1) A(:,1)] = reigen(1,1,E,C,AA) ;
[X(3) A(:,3)] = ieigen(2,1,E,C,AA) ;
X(4) = -X(3) ; A(:,4) = conj(A(:,3)) ;
elseif(e(2)<E & E<(3))
[X(1) A(:,1)]
[X(3) A(:,3)]
[X(5) A(:,5)]
else
[X(1) A(:,1)]
[X(3) A(:,3)]
[X(5) A(:,5)]
end
[X(2) A(:,2)] = reigen(1,0,E,C,AA) ;
[X(5) A(:,5)] = ieigen(3,l,E,C,AA) ;
X(6) = -X(5) ; A(:,6) = conj(A(:,5)) ;
= reigen(1,1,E,C,AA) ;
= reigen(2,1,E,C,AA) ;
= ieigen(3,1,E,C,AA) ;
[X(2) A(:,2)] = reigen(1,0,E,C,AA) ;
[X(4) A(:,4)] = reigen(2,0,E,C,AA) ;
X(6) = -X(5) ; A(:,6) = conj(A(:,5)) ;
= reigen(1,1,E,C,AA) ;
= reigen(2,1,E,C,AA) ;
= reigen(3,1,E,C,AA) ;
[X(2) A(:,2)] = reigen(1,0,E,C,AA) ;
[X(4) A(:,4)] = reigen(2,0,E,C,AA) ;
[X(6) A(:,6)] = reigen(3,0,E,C,AA) ;
DMAT = Dmatrix(C,KH,A,L1,L2,X);
X
V = sqrt(E/RHO)
save PP DMAT A VXH Ll L2
clear
103
% *************************************************************
% GETCONST.M for pp.m
%
% This M-file is to get the material constants of plate
% and calculates components of stiffness matrix.
%
% RHO
= Density
% El, E2
% G12
= Young's modulus
= Shear modulus
% P21, P23 = Poisson's ratio
(following Tsai's convention)
% pi = 3.141592
% *************************************************************
%--- Get material constants.
RHO = 1.52e -6
P21 = 0.31
P23 = 0.54
; % Kg/mm/mm/mm
El = 138.5e-6
E2 = 11.e-6
G12 = 6.3e-6
; % Kg/mm/microsec/microsec
; % These units make a velocity
; % units in mm/microsec.
%--- Calculate components of stiffness matrix.
P12 = P21 *E2/El
C(1,1) = (1-P23*P23)*D*E1
C(3,3) = C(2,2)
C(2,1) = C(1,2)
C(3,1) = C(1,3)
C(3,2) = C(2,3)
C(5,5) = G12
D = 1/(1+P23)/(1-P23-2*P21*P12)
C(2,2) = (1-P21*P12)*D*E2
C(1,2) = P21*(1.+P23)*D*E2
C(1,3) = C(1,2)
C(2,3) = (P23+P21*P12)*D*E2
C(4,4) = (1.-P23-2*P21*P12)*D*E2/2
C(6,6) = G12
THETA = input(' ANGLE OF WAVE PROPAGATION: THETA =')
ANGLE = THETA*pi/180 ; Li = cos(ANGLE) ; L2 = sin(ANGLE)
9
9
;
;
%--- Calculate A matrix components.
AA(1) = C(1,1)*Ll*L1 +C(6,6)*L2*L2 ;AA(2) = C(6,6)*L1 *L1 +C(2,2)*L2*L2;
AA(3) = C(5,5)*Ll*L1 +C(4,4)*L2*L2 ;AA(4) = (C(1,2)+C(6,6))*L1 *L2
;
AA(5) = (C(1,3)+C(5,5))*L1
;AA(6) = (C(2,3)+C(4,4))*L2
,
K . 2*pi
;
KH = K*H
,
H = input(' PLATE THICKNESS H =') ;
104
% *************************************************************
% DETVALUE.M for pp.m
% This M-file calculates the value of boundary condition determinant
% and returns absolute value of it.
% *************************************************************
function y = DETvalue(E,C,AA,KH,THETA)
Ll = cos(THETA*pi/180)
L2 = sin(THETA*pi/180)
A(3,6) = 0
e = sort(eig(Amatrix(0,C,AA)))
if(E <= e(1))
[X(1) A(:,1)] = ieigen(1,1,E,C,AA)
[X(5) A(:,5)] = ieigen(3,1,E,C,AA)
X(4) = -X(3) ; A(:,4) = conj(A(:,3)) ;
elseif(e(1)<E & E<(2))
[X(1) A(:,1)] = reigen(1,1,E,C,AA) ;
[X(3) A(:,3)] = ieigen(2,1,E,C,AA) ;
X(4) = -X(3) ; A(:,4) = conj(A(:,3)) ;
elseif(e(2)<E & E<(3))
[X(1) A(:,1)] = reigen(1,1,E,C,AA) ;
[X(3) A(:,3)] = reigen(2,1,E,C,AA) ;
[X(5) A(:,5)] = ieigen(3,1,E,C,AA) ;
else
[X(1) A(:,1)] = reigen(1,1,E,C,AA) ;
[X(3) A(:,3)] = reigen(2,1,E,C,AA) ;
[X(5) A(:,5)] = reigen(3,1,E,C,AA) ;
end
D = Dmatrix(C,ICH,A,L1,L2,X)
y = abs(det(D));
[X(3) A(:,3)] = ieigen(2,1,E,C,AA) ;
X(2) = -X(1) ; A(:,2) = conj(A(:,1)) ;
X(6) = -X(5) ; A(:,6) = conj(A(:,5)) ;
[X(2) A(:,2)] = reigen(1,0,E,C,AA) ;
[X(5) A(:,5)] = ieigen(3,1,E,C,AA) ;
X(6) = -X(5) ; A(:,6) = conj(A(:,5)) ;
[X(2) A(:,2)] = reigen(1,0,E,C,AA) ;
[X(4) A(:,4)] = reigen(2,0,E,C,AA) ;
X(6) = -X(5) ; A(:,6) = conj(A(:,5)) ;
[X(2) A(:,2)] = reigen(1,0,E,C,AA) ;
[X(4) A(:,4)] = reigen(2,0,E,C,AA) ;
[X(6) A(:,6)] = reigen(3,0,E,C,AA) ;
;
% *************************************************************
% REIGEN.M for pp.m & pd.m
% This M-file is to find real root of L3 and calculate
% corresponding eigen value and eigen vector.
% *************************************************************
function [Y,A] = reigen(K,SIGN,E,C,AA)
LOW = 0
FLOW = value(LOW,K,E,C,AA)
if (SIGN == 1)
HIGH = 5
else
HIGH = -5
end
; % To find positive real root of L3.
; % To find negative real root of L3.
105
FHIGH = value(HIGH,K,E,C,AA)
;
%---- Find root of L3 by half interval search.
index = 1
TOL = le-10
while abs(HIGH-LOW) > TOL
MID = (LOW+HIGH)/2
FMID = value(MID,K,E,C,AA)
ifFMID >0
HIGH = MID
FHIGH = FMID
else
LOW = MID
FLOW = FMID
end
end
%---- Calculate eigen value and eigen vector.
T = Amatrix(LOW,C,AA)
[VECTOR VALUE] = eig(T)
Y = LOW
[VEC,j] = sort(diag(VALUE))
A = VECTOR(:,j(K))
% *************************************************************
% IEIGEN.M for pp.m & pd.m
% This M-file is to find the imaginary root of L3 and calculate
% corresponding eigen value and eigen vector.
% *************************************************************
function [Y,A] = ieigen(K,SIGN,E,C,AA)
%--- Find the negative imaginary root of L3.
i = sqrt(-1)
LOW = -5*i
; % SIGN has no meaning here
FLOW = value(LOW,K,E,C,AA)
HIGH = 0
FHIGH = value(HIGH,K,E,C,AA)
%---- Find root of L3 by half interval search.
index = 1
TOL = le-10
106
while abs(HIGH-LOW) > TOL
MID = (LOW+HIGH)/2
FMID = value(MID,K,E,C,AA)
ifFMID >0
HIGH = MID
FHIGH = FMID
else
LOW = MID
FLOW = FMID
end
end
%---- Calculate eigen value and eigen vector.
T = Amatrix(LOW,C,AA)
[VECTOR VALUE] = eig(T)
Y = LOW
[VEC,j] = sort(diag(VALUE))
A = VECTOR(:,j(K))
% *************************************************************
% VALUE.M
% This M-file returns difference between the assumed eigen value and
% the eigen value corresponding to X.
% When this difference approaches to zero, root of L3 is found.
% *************************************************************
function y = value(X,K,E,C,AA)
Z = Amatrix(X,C,AA)
EIGVALUE = sort(eig(Z))
y = EIGVALUE(K)-E
% *************************************************************
% AMATRIX.M
% This M-file is for component calculation of A matrix.
% *************************************************************
function y=Amatrix(X,C,AA)
A(1,1) = AA(1) +C(5,5)*X*X
A(2,2) = AA(2) +C(4,4)*X*X
A(3,3) = AA(3) +C(3,3)*X*X
107
A(1,2) = AA(4)
A(2,1) = A(1,2)
A(1,3) = AA(5)*X
A(3,1) = A(1,3)
A(2,3) = AA(6)*X
A(3,2) = A(2,3)
y=A
% *************************************************************
% DMATRIX.M for pp.m
% This M-file is to calculate components of boundary condition matrix.
% *************************************************************
function y= Dmat(C,KH,A,L1,L2,X)
D(6,6) = 0
D(1,1)
D(1,2)
D(1,3)
D(1,4)
D(1,5)
D(1,6)
= C(5,5)*(A(1,1)*X(1) +A(3,1)*L1)
= C(5,5)*(A(1,2)*X(2) +A(3,2)*L1)
= C(5,5)*(A(1,3)*X(3) +A(3,3)*L1)
= C(5,5)*(A(1,4)*X(4) +A(3,4) *Ll)
= C(5,5)*(A(1,5)*X(5) +A(3,5)*L1)
= C(5,5)*(A(1,6)*X(6) +A(3,6)*L1)
D(2,1) = C(4,4)*(A(2,1)*X(1) +A(3,1)*L2)
D(2,2) = C(4,4)*(A(2,2)*X(2) +A(3,2)*L2)
D(2,3) = C(4,4)*(A(2,3)*X(3) +A(3,3)*L2)
D(2,4) = C(4,4)*(A(2,4)*X(4) +A(3,4)*L2)
D(2,5) = C(4,4)*(A(2,5)*X(5) +A(3,5)*L2)
D(2,6) = C(4,4)*(A(2,6)*X(6) +A(3,6)*L2)
D(3,1) = C(1,3)*A(1,1)*L1 +C(2,3)*A(2,1)*L2
D(3,2) = C(1,3)*A(1,2)*L1 +C(2,3)*A(2,2)*L2
D(3,3) = C(1,3)*A(1,3)*L1 +C(2,3)*A(2,3)*L2
D(3,4) = C(1,3)*A(1,4)*L1 +C(2,3)*A(2,4)*L2
D(3,5) = C(1,3)*A(1,5)*L1 +C(2,3)*A(2,5)*L2
D(3,6) = C(1,3)*A(1,6)*L1 +C(2,3)*A(2,6)*L2
for j = 1:6
D(4,j) = D(1,j)*exp(-i*KH*X(j))
D(5,j) = D(2,j)*exp(-i*KH*X(j))
D(6,j) = D(3,j)*exp(-i*KH*X(j))
end
y=D
+C(3,3)*A(3,1)*X(1)
+C(3,3)*A(3,2)*X(2)
+C(3,3)*A(3,3)*X(3)
+C(3,3)*A(3,4)*X(4)
+C(3,3)*A(3,5)*X(5)
+C(3,3)*A(3,6)*X(6)
108
% *************************************************************
% PD.M for pp.m
% This M-file is to calculate the value of BCD( boundary condition determinant)
% within an interval selected and show the pattern of BCD.
% After fmding a small interval including root,
% run pp.m to fmd accurate root value.
% *************************************************************
clear
format short e
i = sqrt(-1)
getconst
%-- Get search interval.
A(3,6) = 0
= ')
V1 = input(' V1 (initial)
V2 = input(' V2 (increment) = ')
= ')
V3 = input(' V3 (final)
index =1
%--- Calculate BCD value for each velocity within interval.
for V = V1:V2:V3
E = RHO*V*V
e = sort(eig(Amatrix(0,C,AA)))
if(E <= e(1))
[X(1) A(:,1)] = ieigen(1,1,E,C,AA) ;
[X(5) A(:,5)] = ieigen(3,1,E,C,AA) ;
X(4) = -X(3) ; A(:,4) = conj(A(:,3)) ;
elseif(e(1)<E & E<=e(2))
[X(1) A(:,1)] = reigen(1,1,E,C,AA) ;
[X(3) A(:,3)] = ieigen(2,1,E,C,AA) ;
X(4) = -X(3) ; A(:,4) = conj(A(:,3)) ;
[X(3) A(:,3)] = ieigen(2,1,E,C,AA)
X(2) = -X(1) ; A(:,2) = conj(A(:,1))
X(6) = -X(5) ; A(:,6) = conj(A(:,5))
[X(2) A(:,2)] = reigen(1,0,E,C,AA)
[X(5) A(:,5)] = ieigen(3,1,E,C,AA)
X(6) = -X(5) ; A(:,6) = conj(A(:,5))
elseif(e(2)<E & E<(3))
[X(1) A(:,1)]
[X(3) A(:,3)]
[X(5) A(:,5)]
else
[X(1) A(:,1)]
[X(3) A(:,3)]
[X(5) A(:,5)]
end
= reigen(1,1,E,C,AA) ;
= reigen(2,1,E,C,AA) ;
= ieigen(3,1,E,C,AA) ;
[X(2) A(:,2)] = reigen(1,0,E,C,AA)
[X(4) A(:,4)] = reigen(2,0,E,C,AA)
X(6) = -X(5) ; A(:,6) = conj(A(:,5))
= reigen(1,1,E,C,AA) ;
= reigen(2,1,E,C,AA) ;
= reigen(3,1,E,C,AA) ;
[X(2) A(:,2)] = reigen(1,0,E,C,AA)
[X(4) A(:,4)] = reigen(2,0,E,C,AA)
[X(6) A(:,6)] = reigen(3,0,E,C,AA)
D = Dmatrix(C,ICH,A,L1,L2,X)
V = sqrt(E/RHO)
detl(index) = real(det(D))
det2(index) = imag(det(D))
det3(index) = abs(det(D))
[V detl(index) det2(index) det3(index)]
index = index +1
109
end
V = (V1:V2:V3)
,
%--- Plot value of BCD within interval.
out =[V' detl' det2' I
% plot(V, det1,':',V,det2,'-',V,det3)
% plot(V,detl,V,det2,V,det3)
plot(V,detl,V,det2)
title(' DETERMINANT VALUE OFD MATRIX : THETA = ? : PHI = ? : H = ?')
text(0.5,0.85,'
Real value of det(D) ','sc')
text(0.5,0.8,'
Imag. value of det(D) ','sc')
xlabel(' VELOCITY ( MM / MICRO SEC) ')
ylabel(' DETERMINANT VALUE ')
% *************************************************************
% PU.M for pp.m
%
% This M-file is to calculate and plot displacements of Lamb wave.
% *************************************************************
clear
load PP
%--- Solve for weighting factor W.
for J = 1:5
R(J) = -DMAT(J+1,1)
for K = 1:5
P(J,K) = DMAT(J+1,K+1)
end
end
Q = 'AR'
for J = 1:5
W(J+1) = Q(J)
end
;
W(1)=1
; % Solve PQ = R.
; % W = Weighting factor.
%--- Initialize constants and calculate displacements.
i = sqrt(-1)
XX = 0 ;
N= 1 ;
fl =30
;
YY = 0 ;
WL = 1
f2 = 5
;
,
T=0
,
K = 2*pi/WL ;
for ZZ = 0:-WL/f1:-H
DX(N) = 0 ;
DY(N) = 0 ;
for J = 1:6
DZ(N) = 0 ;
110
DX(N) = DX(N) +W(J)*A(1,J)*exp(i*K*(L1*XX +L2*YY +X(J)*ZZ -V*T));
DY(N) = DY(N) +W(J)*A(2,J)*exp(i*K*(L1*XX +L2*YY +X(J)*ZZ -V*T));
DZ(N) = DZ(N) +W(J)*A(3,J)*exp(i*K*(Ll*XX +L2*YY +X(J)*ZZ -V*1));
end
N = N+1;
end
%--- Plot displacements.
ZZ = (0:- WL/fl: -H)'
;
disp = [DX',DY',conj(DZ')]
% DZ = DZ*(-i)
; % Change of phase
DX = DX*(-i)
,
DY = DY*(-i)
;
% DX = DX/DZ(1)
; % Normalization
% DY = DY/DZ(1)
,
% DZ = DZ/DZ(1)
,
plot(ZZ, DX',':',ZZ,DY','--',ZZ,DZ','-')
title(' X,Y,Z DISPLACEMENT PLOT : H = ?')
text(0.2,0.85,'
X disp. ','sc')
text(0.2,0.8, '- - - - Y disp. ','sc')
text(0.2,0.75,'
Z disp.','sc')
xlabel(' Z / WAVE LENGTH')
ylabel(' DISPLACEMENT ')
111
APPENDIX C
COMPONENTS OF [Ali]
Material stiffness component
contracted form
is used.
> 2
22
4
23, 32 >
has four indices, but usually a
Indices are changed as follows:
*1
11
Ciiki
33
13, 31 > 5
-4 3
12, 21 >
6
Then, components of [Ali] can be written as following for ortho-
tropic materials.
An =
A22
2C56L2L3
2C15L3L1
C221-32 + C441-13 4- 2C261.11,2
2C24L2L3
2C46L3L1
N4412 +
2C341-21/3
2C35L3L1
2 + 2C161,1
C11L + C66122+ C551-3
= C661-11 +
A33 7r
+ 2A-45111-2
2
C45L3
Al2 = A21 = C161.1 + C26L2
(C12 +C66) L11-1
(c-25 + C46) III-3 + (C14 + C56) L3L1
r
.....46L2
T
A13 = A31 =
+ (C36
r,
1,-351/3
ki,-.14
T
1-,11,2
+ C45) 1,2L3 + (Ci3 + C55) L31.1
T
2
Al2 = A21 =
7 2
+
T
2
+ -34-1-13
to
+ `-'25 + C46) L111
+ (C23 + C44)11113 + (C36 + C45) L3L1
For transversly isotropic materials, these can be simplified.
A11
=Ci
+ C661-1
T
A22 = `-'661-1
r,
C55I-3
T
r,
T
+
A33 =
+ C44122 + C33123
=
A 21 = (C12 +C66) L11-2
Al2
A13 = A31 = (C13 + C55) L3Li
Al2 = A 21 = (C23 + C44) 1-21-13
112
APPENDIX D
EXPERIMENTAL WAVEFORMS AND
FFT ANALYSIS OF RAYLEIGH WAVES
40
30
E
20
10
0 .",-^-'`"^-)
10
30
40
50
60
5
5
0
15
10
20
35
30
25
40
TIME ( micro second )
Fig. D.1A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 16.25 mm thickness
(0=00).
x105
5
1
3
4
5
6
FREQUENCY ( MHz )
Fig. D.1.8 Frequency analysis of Fig. D.1.A.
7
113
20
10
10
20
30
40
50
5
0
5
10
15
TIME (
20
25
30
35
40
micro second )
Fig. D.2.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 16.25 mm thickness
( 0 = 20° ).
x105
6
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. D.2.B Frequency analysis of Fig. D.2.A.
7
114
30
20
10
0 L'irVA
10
20
30
40
50
5
0
5
10
15
20
30
25
35
40
TIME ( micro second )
Fig. D.3.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 16.25 mm thickness
( 0 = 50° ).
x105
16
14
12 -
10-
86-
420
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. D.3.B Frequency analysis of Fig. D.3.A.
7
115
30
20
10
cy\
- 10
- 20
- 30
- 40
- 50
60
-10
-5
0
5
10
15
20
25
30
35
40
TIME ( micro second )
Fig. D.4.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 16.25 mm thickness
( 0 = 70° ).
x105
12
10
1
oo
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. D.4.B Frequency analysis of Fig. D.4.A.
7
116
40
20
?:*.
0
4i
-tt
a
c -20
E-.
EM.
E-'
o
-40
Ix
c.i
0
00
-60
-4
-80
z
-100
-10
-5
0
5
10
20
15
25
30
35
40
TIME ( micro second )
Fig. D.5.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 16.25 mm thickness
( 0 = 90° ).
x106
6
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. D.5.B Frequency analysis of Fig. D.5.A.
7
117
APPENDIX E
EXPERIMENTAL WAVEFORMS AND
FFT ANALYSIS OF LAMB WAVES
150
100 E
50 -
50
100 -
150
5
0
10
5
15
35
30
25
20
40
TIME ( micro second )
Fig. E.1.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
( 0 = 0° ).
x1013
12
10
8
6
4
2
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. E.1.B Frequency analysis of Fig. E.1 A
7
118
60
40
20
Ar
-20
40 60 -
80 100
5
0
5
10
15
TIME (
20
25
30
35
40
micro second )
Fig. E.2.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
( 0 = 20° ).
x105
FREQUENCY ( MHz )
Fig. E.2.8 Frequency analysis of Fig. E.2.A.
119
80
60
40
20
T-NN/0-nn,-
20
40
5
0
5
10
15
20
25
30
35
40
TIME ( micro second )
Fig. E.3.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
(0=40°).
x100
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. E.3.B Frequency analysis of Fig. E.3.A.
7
120
50
40
30
44
20
10
,-.1.(11J-111.4
0
-10
20
Z 30
40
50
5
0
5
10
15
20
25
30
35
40
TIME ( micro second )
Fig. E.4.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
( 9 = 50° ).
x105
16
14
12
10
8
6
4J
2
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. E.4.B Frequency analysis of Fig. E.4.A.
7
121
60
40
E
20
0 7,*.nrvrn.on
f,
i
20
i
40 -
60
5
0
10
5
15
20
30
25
35
40
TIME ( micro second )
Fig. E.5.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
( 0 = 60° ).
x106
1.8
1.6 1.4 -
0.8
0.6
t'
0.4
0.2
Oo
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. E.5.B Frequency analysis of Fig. E.5.A.
7
122
80
...-.
60
40
20
-20
-40
1
-60
-80
-5
0
5
10
15
20
25
30
35
40
TIME ( micro second )
Fig. E.6.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
( 0 = 80° ).
4
x106
3.5
2.5
1.5
0.5
=
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. E.6.B Frequency analysis of Fig. E.6.A.
7
123
80
60 -
40
20
1
20
40
5
0
5
15
10
20
30
25
35
40
TIME ( micro second )
Fig. E.7.A Laser generated ultrasonic waves in the unidirectional
graphite/epoxy composite plate of 0.87 mm thickness
( 0 = 90° ).
X105
14
12
10
1
2
3
4
5
6
FREQUENCY ( MHz )
Fig. E.7.B Frequency analysis of Fig. E.7.A.
7